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Talk:Generating function

2024-02-19T13:55:37Z

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==References please==
Please give the references for the very nice formulas in the section on asymptotics of coefficients.
[[User:Asympt|Asympt]] ([[User talk:Asympt|talk]]) 18:57, 21 November 2021 (UTC)
:I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő, ''Problems and Theorems in Analysis, Vol 1.'' (1972), Exercise 174. And I see a citation to Wilf ''generatingfunctionology'' (1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 22:56, 21 November 2021 (UTC)

==Ancient comment==

The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
-[[User:Iopq|Iopq]] 19:59, 18 October 2005 (UTC)

== Definition ==

I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:

"A generation function is a transformation that converts a given sequence, ''S = {an}'', into a continous function, f(x), through a series expantion whose coeficients are the elements ''an'' of the sequence ''S''."

or something similar you find more appropiate.

:Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity. [[User:Charles Matthews|Charles Matthews]] 12:19, 16 November 2005 (UTC)

: I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the ''formal'' nature of the series. --[[User:Zero0000|Zero]] 22:52, 16 November 2005 (UTC)

::Absolutely. (No pun intended.) To call these things "continuous" is absurd. [[User:Michael Hardy|Michael Hardy]] 20:13, 17 November 2005 (UTC)

:::Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.[[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 14:45, 15 May 2019 (UTC)

:::: Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and [https://arxiv.org/abs/1106.5480 here] is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also [https://math.mit.edu/~rstan/ec/ec1.pdf EC1], notes on Chapter 1. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 15:20, 15 May 2019 (UTC)

::::: I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like <math>\Psi(x)</math> and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject. [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:54, 15 May 2019 (UTC)

I just noticed that the german version is not liked here it's called "Erzeugende Funktionen", url is here: http://de.wikipedia.org/wiki/Erzeugende_Funktion — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.254.173.61|80.254.173.61]] ([[User talk:80.254.173.61#top|talk]]) 16:44, 18 December 2005 (UTC)

== Examples please! ==

''In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. '''For example,'''...'' (a nice easy example or two, please!)

This article is fairly typical of current Wikipedia mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!

Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.

--[[User:84.9.78.198|84.9.78.198]] 14:14, 27 November 2006 (UTC)

:If you read on past the ''Definitions'' section you will find an ''Examples'' section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the [[Fibonacci number]]s is derived. If ''Examples'' came before ''Definitions'' the article would be more difficult to follow, as you would not know what the ''Examples'' were meant to be illustrating. [[User:Gandalf61|Gandalf61]] 14:41, 27 November 2006 (UTC)

== Uniqueness of F ==

I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-''X''). If ''F'' is any ring with a unit, not necessarily commutative or an integral domain, then the only power series <math>f(X) \in F[[X]]</math> such that <math>1=(1-X)f(X)</math> is <math>f(X)=1+X+X^2+\dots</math>. To see this, let <math> f(X) = f_0 + f_1 X + f_2 X^2 + \dots</math>. Then <math>(1-X)f(X) = f_0 + (f_1 - f_0) X + (f_2 - f_1) X^2 + ...</math>. Solving <math> 1 = (1-X)f(X) </math> means that <math> f_0 = 1, (f_1 - f_0) = 0, (f_2 - f_0) = 0, \dots</math> and therefore <math> 1 = f_0 = f_1 = f_2 = \dots </math>. The only multiplication in the ring ''F'' is used in this proof is multiplication by 1 and -1 in ''F''. Therefore neither general commutativity of the ring ''F'' nor ''F'' being an integral domain is required. Indeed multiplication in ''F'' need not even be associative. (So, the result holds if ''F'' is the octonions, for example.) It is only required that multiplication in ''F'' have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by ''F'' being a ring. Of course, multiplication by ''X'' in <math>F[[X]]</math> has been used, and commutativity of this operation , that is, <math>Xa = aX</math> has been used, as has the fact if <math>Xa=0</math> then <math>a=0</math>. In general, though if the ring of coefficient is not an integral domain or commutative, then neither is the resulting power series ring. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]] • [[Special:Contributions/DRLB|contribs]]) 15:26, 17 October 2008 (UTC) 

: That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --[[User:Charleyc|Charleyc]] ([[User talk:Charleyc|talk]]) 16:23, 18 October 2008 (UTC)

:: Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the article.) [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]]) 14:50, 20 October 2008 (UTC)

== Formulae ==

I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3... —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.187.35.160|85.187.35.160]] ([[User talk:85.187.35.160|talk]]) 13:47, 20 August 2009 (UTC) 

:Fixed - I have replaced <math>\sum_{n\in\mathbf{N}}</math> with <math>\sum_{n=0}^{\infty}</math>, which was clearly what was intended in each case. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:55, 20 August 2009 (UTC)

== "Generating series" terminology ==

The [http://en.wikipedia.org/w/index.php?title=Generating_function&oldid=364447959 current version] of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"? [[User:Quantling|Quantling]] ([[User talk:Quantling|talk]]) 16:00, 27 May 2010 (UTC)

:The "series" in "generating series" refers to [[formal power series]], where convergence is not much of an issue either (the term "generating formal power series" would be a bit heavy). Series are not necessarily about convergence, so I don't think this is much of a problem. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:35, 28 May 2010 (UTC)

== Is this a generating function? ==

<math>

\pi(\cot (\pi(c+z))-2\cot (2\pi(c-z))-2\cot (2\pi(c+z))+\cot (\pi(c-z)))

= -2 \left ( \sum_{k=0}^\infty z^{2k} \sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{2k+1}} - \frac{1}{(x-c-1/2)^{2k+1}} \right )

</math>

taking multiple derivatives with respect to z closed form sums can be obtained such as:

<math>

\pi^{3}(8(\cot(2c\pi)+\cot^{3}(2c\pi))-(\cot(c\pi)+\cot^{3}(c\pi))) =\sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{3}} - \frac{1}{(x-c-1/2)^{3}}

</math>

<math>

= -\frac{\pi^{3}\sin(c\pi)}{\cos^{3}(c\pi)}

</math>

<math>

\cot(\pi(c+z)) \approx \cot(c\pi)-z\pi(1+\cot^{2}(c\pi))+z^{2}\pi^{2}(\cot(c\pi)+\cot^{3}(c\pi))

</math>

http://iamned.com/math —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.161.40.148|67.161.40.148]] ([[User talk:67.161.40.148|talk]]) 11:02, 30 June 2010 (UTC) 

== confusing definition ==

The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing. [[User:RobLandau|RobLandau]] ([[User talk:RobLandau|talk]]) 07:40, 8 February 2018 (UTC)
:{{ping|RobLandau}} Generating functions are ''not'' sequences. The article does not say that; the article says they are used to ''describe'' sequences. I see no confusion here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:44, 8 February 2018 (UTC)
:: The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 13:05, 8 February 2018 (UTC)

Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which {{ping|Joel B. Lewis}} has twice restored, doesn’t make sense. The original and restored sentence ''The sum of this infinite series is the generating function'', as I said in my edit summary when I changed it and as RobLandau said above, is certain to give some people the impression that it means “The number that this series sums to is the generating function”, which is not right.

My replacement sentence, which I’m not wedded to, said ''The summation of this infinite series is the generating function''. Here ''[[summation]]'', as per its article, means ''the addition of a sequence of numbers'', which correctly refers to the entity rather than the result.

Joel restored the original with the edit summary ''The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)'' But many readers will not understand that here ''sum'' is intended to mean ''the whole infinite series''. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both ''sum'' and ''summation''. Thanks! [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 16:16, 29 May 2018 (UTC)

:I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using, ''This [[formal power series]] is the generating function.'' --[[User:Wcherowi|Bill Cherowitzo ]] ([[User talk:Wcherowi|talk]]) 17:04, 29 May 2018 (UTC)

:: Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is [https://en.wikipedia.org/w/index.php?title=Generating_function&diff=824616355&oldid=814318669 this one], which did not "restore" anything. The phrase "the summation of a series" makes no sense; if it did make sense, it would mean exactly the same as "the sum of the series". Indeed, the series ''is'' the sum; this sum is not a [real or complex] number because the individual summands are not [real or complex] numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:10, 29 May 2018 (UTC)
:::Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 19:31, 29 May 2018 (UTC)

:::: Your comment describes me has having "twice restored" something. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 29 May 2018 (UTC)

:::::Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 20:38, 29 May 2018 (UTC)
== Function* listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px|left]]
An editor has asked for a discussion to address the redirect [[Function*]]. Please participate in [[Wikipedia:Redirects for discussion/Log/2019 May 11#Function*|the redirect discussion]] if you wish to do so.  [[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:51, 11 May 2019 (UTC)

== Article is a mess ==

This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
* By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for Lambert, Bell, and formal Dirichlet series). The current version is trying to do ''way'' too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
* The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
* There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments".... [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:31, 15 May 2019 (UTC)

== must it be infinite? ==

Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial <math>(\frac{2}{5}x^1 + \frac{1}{5}x^2 + \frac{1}{5}x^3 + \frac{1}{5}x^4)^{64} </math>. Is that not a generating function because it's not infinite? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 14:56, 16 October 2019 (UTC)
: Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:17, 16 October 2019 (UTC)
:: The wiki-linking in the lede is also rather [[WP:SUBMARINE|submarine]]. It links to [[formal power series]] with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to [[formal power series]] ''again'' with the text "formal series". Next we get {{tq|Generating functions were first introduced by Abraham de Moivre in 1730}} — fine — {{tq| in order to solve the general linear recurrence problem.}} Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence problem just the challenge of understanding linear recurrences in general? [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 05:21, 17 October 2019 (UTC)

== Formula for generating function for a linear recursive sequrnce. ==

The following formula is really easy to use. Shall it be included in this article?

Let <math>s_n</math> be a linear recursive sequence of order k with initial conditions
<math> \{s_0, s_1, \ldots, s_{k-1}\}</math> and recursive relation
<math>s_n = \sum_{i=1}^k a_i s_{n-i}.</math>

Then the generating function for $s_n$ is given by the formula

<math>(\sum_{i=0}^{k-1} ( \sum_{j=0}^{i} (-a_j)* s_{i-j}) * x^{i-k})/f(x^{-1})</math> — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Kaiwang45|Kaiwang45]] ([[User talk:Kaiwang45#top|talk]] • [[Special:Contributions/Kaiwang45|contribs]]) 15:49, 27 July 2020 (UTC) 

== Blackboard bold formatting ==

{{reply|Quantling}} Greetings! Regarding [https://en.wikipedia.org/w/index.php?title=Generating_function&oldid=prev&diff=1146738276&diffmode=source this revert]...the use of {{tag|math}} is required by [[MOS:BBB]]. If we want the nearby markup to be consistent, that's fine; we would just need to convert it to also use {{tag|math}}. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 16:21, 27 March 2023 (UTC)
:{{reply to|Beland}} Good point. To be more consistent with [[MOS:STYLERET]], other possibilities are to use
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'', ''b'' ∈ '''N'''}}, {{math|''a'' ≥ 2}}, and {{math|0 ≤ ''b'' < ''a''}}.
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}} and {{math|0 ≤ ''b'' < ''a''}} are integers.
:What do you think? —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:39, 27 March 2023 (UTC)
::{{reply|Quantling}} "{{math|''a''}} and {{math|''b''}} are integers" is certainly a lot less jargony than using the blackboard bold notation. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 17:45, 27 March 2023 (UTC)
:::I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:55, 27 March 2023 (UTC)
::::Done; thanks for your help ironing this out! -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 22:09, 27 March 2023 (UTC)

== Remove Sections ==

It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?

For starters, we should probably remove P-holonomic functions and J-fractions and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.

Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! (Ex: The 'Article is a mess' post above seems rather insightful, and I'll try to propose concrete edits which might circumvent the proposed issues.)

Also, how about this one: We just list a couple applications of generating functions (I honestly think snake oil or something is a good enough thing to convince people that they're 'useful', and then maintain a 'main article' on applications). I wish to scrap off the entire J-fraction part, write something about them in a main link, write about transforming between ordinary and exponential generating functions and then remove the whole transforming part.

[[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 03:27, 25 November 2023 (UTC)

:In hindsight, the applications part can be cut down here and there. However, it's the ordinary generating functions part that needs to basically go out of the window. It's WAY too extensive. [[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 06:26, 16 February 2024 (UTC)
:Spreading the content to multiple articles that specialize in specific aspects sounds good to me. However, much of the content is quite interesting to me, so I am hoping that, as it leaves this article, it does go somewhere, not merely to the [[null device]]. —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 13:54, 16 February 2024 (UTC)
::We could start by checking that all the content on ordinary generating functions makes it to an ordinary generating functions wiki, and then we can prune most of the content from here. [[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 13:55, 19 February 2024 (UTC)

Talk:Generating function

2024-02-16T06:26:28Z

Yeetcode: /* Remove Sections */ Reply

Wikipedia:WikiProject Mathematics/Participants

2024-01-07T09:51:10Z

Yeetcode: /* Active participants U–Z */

{{WikiProject mathematics tabs}}
The following table lists users who are participants in the [[Wikipedia:WikiProject Mathematics|WikiProject Mathematics]]. Please feel free to add yourself to the list. Instructions can be displayed by editing the page (if you can't figure out how to edit the table, just leave your info at the bottom of the page and someone will probably update the table for you).

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__TOC__
==Active participants A–E==

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
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|-
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||
|-
|[[User:AfroThundr3007730|AfroThundr3007730]] ([[User_talk:AfroThundr3007730|T]] [[Special:Contributions/AfroThundr3007730|C]])
||[[Calculus]], [[Physics]], [[Logic]], [[Mathematics]]
|| Nothing special, just a general love of math.
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|[[User:AGK|AGK]] ([[User_talk:AGK|T]] [[Special:Contributions/AGK|C]]) ||[[Algebraic geometry]], various [[calculus]] topics ([[differential calculus]] in particular) ||
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|[[User:Aiden Fisher|Aiden Fisher]] ([[User_talk:Aiden Fisher|T]] [[Special:Contributions/Aiden Fisher|C]] Old([[User_talk:Donmegapoppadoc|T]] [[Special:Contributions/Donmegapoppadoc|C]]))|| [[Stochastic process]], [[Markov process|Markov processes]], [[Queueing theory]] ||I'm a PhD candidate at the University of Adelaide
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|[[User:Akihironihongo|Akihironihongo]] ([[User_talk:Akihironihongo|T]] [[Special:Contributions/Akihironihongo|C]]) || Anything and everything pertaining to mathematics || A High School Sophomore taking College-level mathematics courses.
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|[[User:Akvilas|Akvilas]] ([[User_talk:Akvilas|T]] [[Special:Contributions/Akvilas|C]]) ||[[Numerical Analysis]], [[Fluid Dynamics]], [[Physics]] || Bachelor, heading towards PhD in applied mathematics.
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|[[User:A legend|A legend]] ([[User_talk:A legend|T]] [[Special:Contributions/A legend|C]]) || Abstract Algebra || PDE's drive me CRAZY!
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|[[User:Alex Bakharev|Alex Bakharev]] ([[User_talk:Alex Bakharev|T]] [[Special:Contributions/Alex Bakharev|C]]) || [[Numerical analysis]], applications to [[Rheology]] and [[Fluid Dynamics]], elementary mathematics, biographies || I am an R&D engineer, developing models and writing codes for the plastic industry.
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|[[User:Alterationx10 | Alterationx10]]  [[]]) || || I am a Ph.D. student in noncommutative geometry at the Max Planck Institute in Bonn.
|-
|[[User:AHEJJWILEMAMALIDGED|AHEJJWILEMAMALIDGED]] ([[User talk:AHEJJWILEMAMALIDGED|talk]]
||All mathematics Included[[Further Mathematics]],[[Spatial geometry]],[[Area]],[[Volume]],[[Integral]],[[Differential geometry]],[[Calculus]],[[Discrete mathematics]]
||I have acquired information in the fields of mathematics and am a scholar of them
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|[[User:Ammarpad|Ammarpad]] ([[User talk:Ammarpad|talk]] [[Special:Contributions/Ammarpad|C]])
||[[Permutation]]s, [[Calculus]]
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|[[User:An Unidentified Martian|An Unidentified Martian]] ([[User_talk:An Unidentified Martian|T]] [[Special:Contributions/An Unidentified Martian|C]]) || [[Geometry]], [[Infinite Series]], [[Number Theory]], [[Calculus]], [[Chaos Theory]] || A maths enthusiast who is quite competitive.
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|[[User:Anand QED|Anand]] ([[User_talk:Anand QED|T]] [[Special:Contributions/Anand QED|C]])|| [[Differential geometry]], [[Mathematical Analysis]], [[Abstract algebra]], [[Calculus]], [[Multivariable Calculus]], ||Neither academic nor talented. Know my way around pure mathematics. Bourbaki fan.
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|[[User:Andrewa | Andrewa]] ([[User_talk:Andrewa|T]] [[Special:Contributions/Andrewa|C]])||[[model theory]], [[non-standard analysis]], [[history of mathematics]], [[philosophy of mathematics]] || Majors in [[pure mathematics]] and [[philosophy]], particularly [[formal logic]], [[metalogic]] and [[meaning and reference]]. Interested in making mathematics and logic articles accessible to non-mathematicians and even to mathematicians with other areas of specialisation.
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|[[User:Andrewmc123|Andrewmc123]] ([[User_talk:Andrewmc123|T |]] [[Special:Contributions/Andrewmc123|C]]) || Interested in most things to do with Mathematics. Expert in alegra and algebraic geometry || I have acheived the Scottish Advanced Higher Mathematics.
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|[[User:AndyrooP|AndyrooP]] ([[User_talk:AndyrooP|T]] [[Special:Contributions/AndyrooP|C]])|| [[Geometry]], [[Topology]], [[Mathematical Physics]] || I am a PhD student at [[UNC Chapel Hill]]. I completed my undergraduate degree in mathematics at [[UC San Diego]].
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|[[User:AnnekeBart|AnnekeBart]] ([[User_talk:AnnekeBart|T |]] [[Special:Contributions/AnnekeBart|C]]) || [[low-dimensional topology]] || My research is in (geometric) topology. I'm also interested in the history of mathematics and ancient Egypt.
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|[[User:AnthonyMarkes|AnthonyMarkes]] ([[User_talk:AnthonyMarkes|T |]] [[Special:Contributions/AnthonyMarkes|C]]) || [[Pure Mathematics]], [[Physics]], [[Cryptology]], [[Triangle Optimization]] || I am primarily interested in mathematics as it applies to engineering, motion, energy and programming.
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|[[User:Anti-Quasar|Anti-Quasar]] ([[User_talk:Anti-Quasar|T |]] [[Special:Contributions/Anti-Quasar|C]]) || [[Mathematical Physics]], [[Astrophysics]] || The [[Development]] of the [http://en.wikipedia.org/wiki/Mathematical_physics Mathematical Methods] for application to [[problems]] in the [[Physics]], [[Space]] and [[Motion]].
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|[[User:arcfrk|arcfrk]] ([[User_talk:arcfrk|T]] [[Special:Contributions/arcfrk|C]]) || [[abstract algebra|algebra]], [[combinatorics]], [[geometry]], [[history of mathematics]], [[Lie theory]], [[number theory]] || Mathematical interests are fairly diverse, prefer expository style ('big picture') to technicalities, expert in [[representation theory]].
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|[[User:ArkianNWM|ArkianNWM]] ([[User_talk:ArkianNWM|T]] [[Special:Contributions/ArkianNWM|C]]) || Interested in most aspects of mathematics, though I am particularly interested in [[Mathematical analysis|analysis]] and [[calculus]], [[series (mathematics)|series]], and [[numeral systems]]. || I am an undergraduate seeking a BS in [[materials science]] and one mathematics at the [[University of Arizona]]
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|[[User:Arthur_Rubin|Arthur Rubin]] ([[User_talk:Arthur_Rubin|T]] [[Special:Contributions/Arthur_Rubin|C]]) || [[mathematical logic]] || (You want only mathematical interests here, right.) Ph.D. in mathematics from [[Caltech]], with a specialization in mathematical logic. Currently working for an aerospace company, where I '''do''' make use of advanced mathematics, believe it or not.
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|[[User:alphaomega|alphaomega]] ([[User_talk:alphaomega|T]] [[Special:Contributions/alphaomega|C]]) || [[numeral analysis]], [[mathematical logic]], [[topology]], [[combinatorics]], [[functional analysis]] and [[mathematical logic]], [[mathematical physics]] || Ph.D. in Mathematics and Electrical Engineering
|-

|[[User:Artie_p|Artie_p]] ([[User_talk:Artie_p|T]] [[Special:Contributions/Artie_p|C]]) || [[algebra]],[[algebraic topology]],[[algebraic geometry]] ||I'm a Ph.D. student, working in algebraic geometry and topology.
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|[[User:Ateeq.s|Ateeq Sharfuddin]] ([[User_talk:Ateeq.s|T]] [[Special:Contributions/Ateeq.s|C]]) || number theory; linear algebra; abstract algebra; real analysis; probability theory || Boo!
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|[[User:Avraham|Avi]] ([[User_talk:Avraham|T]] [[Special:Contributions/Avraham|C]]) || [[applied mathematics]], [[probability and statistics]], [[actuarial science]] || [[Casualty insurance|Casualty]] [[actuary]].
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|[[User:AxelBoldt | AxelBoldt]] ([[User_talk:AxelBoldt|T]] [[Special:Contributions/AxelBoldt|C]])|| ||
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|[[User:Ashraf isslam|Ashraf isslam]] ([[User_talk:Ashraf isslam|T]] [[Special:Contributions/Ashraf isslam|C]]) || [[Applied Mathematics]], [[Probability]], [[Statistics]], [[Algebra]], [[Calculus]], [[Geometry]], [[Trigonometry]], [[Mathematical Economics]] || Undergraduate student from Bangladesh. Extremely interested in Mathematics. Participated in many Math Olympiads.
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|[[User: ASimpleCompanion|ASimpleCompanion]] ([[User_talk:ASimpleCompanion|t]] [[Special:Contributions/ASimpleCompanion|C]]) ||History of math||[[Euclid]] is amazing!!
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|[[User:Barak Sh|Barak Sh]] ([[User_talk:Barak Sh|T]] [[Special:Contributions/Barak Sh|C]]) || Mainly [[Mathematical Physics]] || Currently working on a B.Sc in Mathematics & Physics at [[Tel Aviv University]].
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|[[User:Bci2|Bci2]] ([[User_talk:Bci2|T]] [[Special:Contributions/Bci2|C]]) || Mainly [[Mathematical Physics, Category Theory, HDA and Mathematical Biology]] || M.Sc. in and Mathematical Biology and PhD in Physics, now at [[University of Illinois]].
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|[[User:BertieOnestone|Bertie Onestone]] ([[User_talk:BertieOnestone|T]] [[Special:Contributions/BertieOnestone|C]]) || [[Stochastic calculus]] || Actively involved in academic research.
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|[[User:Billlion | Billlion]] ([[User_talk:Billlion|T]] [[Special:Contributions/Billlion|C]])|| [[Inverse problem]]s, [[Differential geometry]]||
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|[[User:Bharath628 | Bharath]]   ([[User_talk:Bharath628|T]]  [[Special:Contributions/Bharath628|C]] )||
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|[[User:Bjcairns | Ben Cairns]] ([[User_talk:Bjcairns|T]] [[Special:Contributions/Bjcairns|C]])|| [[probability theory]], [[applied probability]], [[stochastic processes]], [[statistics]], [[Mathematical analysis|analysis]] || I'm a PhD student in applied probability and stochastic processes. Most of my mathematical interest (and expertise) lies in these fields and in related topics such as analysis, a little measure theory, and so on.
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|[[User:BK Drinkwater|BK Drinkwater]] ([[User_talk:BK Drinkwater|T]] [[Special:Contributions/BK Drinkwater|C]]) || logic, synthetic geometry, recursion theory || New to the wiki. Will help out where I can.
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|[[User:bensculfor|Ben Sculfor]] ([[User_talk:bensculfor|T]] [[Special:Contributions/bensculfor|C]])
|| [[algebraic geometry]] and [[algebraic groups]], in particular groups over the [[field with one element]] || MSci (Hons) in maths, hoping to go back and complete a PhD someday.
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|[[User:Ben Tillman|Ben Tillman]] ([[User_talk:Ben Tillman|T]] [[Special:Contributions/Ben Tillman|C]])|| [[Linear algebra]], [[Functional analysis]], [[Complex analysis]], [[Group theory]], [[Topological group]]s, [[History of mathematics]] || B Math (Hons), B Comp Sci so far. Many areas of mathematics interest me, but I'm not an expert in any particular area.
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|[[User:blake.boursaw|Blake Boursaw]] ([[User_talk:blake.boursaw|T]] [[Special:Contributions/blake.boursaw|C]]) || [[Topological dynamics]], [[Geometric Group Theory]] || I'm a pure mathematician by training and, currently, an applied statistician working in health policy by trade. Accordingly, I have an embarrassingly wide variety of interests.
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|[[User:Bo Jacoby|Bo Jacoby]] ([[User_talk:Bo Jacoby|T]] [[Special:Contributions/Bo Jacoby|C]])|| [[inferential statistics]], [[exponentiation]], [[root of unity]], [[Durand-Kerner method]], [[Multiset]] || I studied physics and mathematics long ago.
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|[[User:Borisblue|Borisblue]] ([[User_talk:Borisblue|T]] [[Special:Contributions/Borisblue|C]])|| Nominated [[Isaac Newton]], [[Carl Friedrich Gauss]] and [[Leonhard Euler]] to Featured status, so I guess my specialty is math biographies || Undergrad.
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|[[User:Blnguyen| Blnguyen]] ([[User_talk:Blnguyen|T]] [[Special:Contributions/Blnguyen|C]])|| [[algebra]] [[Hopf algebra]] || PhD student in theoretical physics. Hons B.Sc last year in math physics - [[Quantum groups]] and [[Statistical mechanics]] was my project - although that wasn't my strong point
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|[[User:BradBeattie| Brad Beattie]] ([[User_talk:BradBeattie|T]] [[Special:Contributions/BradBeattie|C]])|| [[Analysis_of_algorithms|algorithm analysis]], [[Computational_complexity_theory|complexity]], [[graph theory]], [[cryptography]], [[combinatorics]] || Finished my BMath a couple years ago. Did my honours in CompSci with a minor in C&O.
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|[[User:Brad7777|Brad]] ([[User_talk:Brad7777|T]] [[Special:Contributions/Brad7777|C]]) || The existence of mathematics. The scope of mathematics. Mathematical abstraction ||
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|[[User:Brent Perreault|Brent]] ([[User_talk:Brent_Perreault|T]] [[Special:Contributions/Brent_Perreault|C]]) || [[Kalman filter]] and [[matrix calculus]] as well as math articles related to physics. || I just finished Bachelors in math and physics, and I'm now going to grad school for physics.
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|[[User:Bte99|bte99]] ([[User_talk:Bte99|T]] [[Special:Contributions/Bte99|C]]) || [[Algebra]], [[Statistics]] || none (yet)
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|[[User:BurkeFT|BurkeFT]] ([[User_talk:BurkeFT|T]] [[Special:Contributions/BurkeFT|C]]) || [[Logic]], [[Foundations of mathematics]], [[Geometry]] || Philosophy PhD, Stanford.
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|[[User:Cabellwg|Cabellwg]] ([[User_talk:Cabellwg|T]] [[Special:Contributions/Cabellwg|C]])
|| [[group theory]], [[number theory]], [[category theory]]
|| B.S. student in mathematics
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|[[User:Can You Prove That You're Human|Can You Prove That You're Human]] ([[User_talk:Can You Prove That You're Human|T]] [[Special:Contributions/Can You Prove That You're Human|C]]) || [[Algebra]]; [[Calculus]]; [[Vector (mathematics and physics)|Vectors]]; [[Trigonometry]]; [[Mathematical functions|Mathematical Functions]]; [[Economics|Mathematical Analysis in Economics]]|| Working toward Honours Bachelor's Degree in Economics
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|[[User:CarnivorousBunny|CarnivorousBunny]] ([[User_talk:CarnivorousBunny|T]] [[Special:Contributions/CarnivorousBunny|C]])
|| [[philosophy of mathematics]], [[logic]], [[geometry]], [[calculus]], [[game theory]], [[knot theory]]
|| Idle pursuit.
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|[[User:Caterpillar_tree|Caterpillar_tree]] ([[User_talk:Caterpillar_tree|T]] [[Special:Contributions/Caterpillar_tree|C]]) || [[Graph theory]], [[Universal algebra]], [[Algebraic topology]] || I'm not ready to metamorphose into a path yet.
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|[[User:catsarefurrytheory|catsarefurrytheory]] ([[catsarefurrytheory|T]] [[Special:Contributions/catsarefurrytheory|C]])|| [[Number Theory]], [[Category Theory]], [[Graph Theory]]|| Computer scientist with mathematical bent
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|[[User:Cayl2357|Cayl2357]] ([[User_talk:Cayl2357|T]] [[Special:Contributions/Cayl2357|C]])|| [[Analysis]], [[Graph Theory]], [[Probability]] || B.S. in Math, continuing studies in applications of graph theory
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|[[User:C0N6R355|C0N6R355]] ([[User_talk:C0N6R355|T]] [[Special:Contributions/C0N6R355|C]]) || [[fractal]]s, [[mobius strip]]s, [[fourth dimension]].|| Programmer by trade, C++, delved into number theory.
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|[[User:CBM|CBM]] ([[User_talk:CBM|T]] [[Special:Contributions/CBM|C]])
||[[mathematical logic]]
||
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|[[User:CDicken|CDicken]] ([[User_talk:CDicken|T]] [[Special:Contributions/CDicken|C]]) || Pure Maths, mainly number theory || Maths, Further Maths, Physics, Chemistry and Electronics A level Student, and I am planning to do a BSc in Maths afterwards. Without a doubt i'm not as good as some of the people on here, but i will do what i can.
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|[[User:Cegalegolog99|Cega]][[User:Cegalegolog99|'''''LEGO''''']][[User:Cegalegolog99|log]][[User:Cegalegolog99|'''9'''9!]]||All math||I am doing really advanced math, and I love it!:-)
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|[[User:Chaoticallyc]] ([[User_talk:Chaoticallyc|T]] [[Special:Contributions/Chaoticallyc|C]])||Most areas of mathematics || I love mathematics and am currently studying math and computer science in University. I'm just starting putting my knowledge to use on Wikipedia.
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|[[User:Charles Matthews | Charles Matthews]] ([[User_talk:Charles Matthews|T]] [[Special:Contributions/Charles Matthews|C]])||Worked in [[number theory]] once upon a time; I write here over the whole range || I've added about 450 mathematics topic pages, many biographies, and lists of mathematical topics. I'm an admin and arbitrator here.
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|[[User:Chalst | Charles Stewart]] ([[User_talk:Chalst|T]] [[Special:Contributions/Chalst|C]])||[[mathematical logic]], [[category theory]], mathematical structures for semantics (eg. [[set theory]], [[Boolean algebra]]s, [[domain theory]]) || I'm a logician and I'm mostly here for the overlap with logic in [[mathematical logic]], but I've some other interests in mathematics here and there.
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|[[User:Chlorinee|Chlorinee]] || [[Graph Theory]],[[Combinatorics]] || A person who does some math and thinks about domino tilings on periodic lattices sometimes.
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|[[User:Count Von Aubel|Count Von Aubel]] || [[Geometry]] || Mostly interested in synthetic geometry.
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|[[User:Cpatra1984|Chinmoy]] ([[User_talk:Cpatra1984|T]] [[Special:Contributions/patra1984|C]]) || [[Algebra]], [[Geometry]], [[Functional Analysis]], [[Topology]]|| I am Assistant Professor in Mathematics.
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|[[User:Cholpon.tuzabaeva|Cholpon Tuzabaeva]] ([[User_talk:Cholpon.tuzabaeva|T]] [[Special:Contributions/Cholpon.tuzabaeva|C]]) || All mathematics topics || I am only in secondary school but love math and think I can help by adding links to articles that helped me understand the topics
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|[[User:Hillman|Chris Hillman]] ([[User_talk:Hillman|T]] [[Special:Contributions/Hillman|C]]) || [[mathematics]] || Broad interests but (currently) narrow focus
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|[[User:Cirdan90|Cirdan90]] ([[User_talk:Cirdan90|T]] [[Special:Contributions/Cirdan90|C]]) || Numerical Linear Algebra, Computational Mathematics || Research Fellow at [[Università di Pisa]]
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|[[User:jcherman|Jaken Herman]] ([[User_talk:jcherman|T]] [[Special:Contributions/jcherman|C]]) || [[Artificial Neural Networks]], [[Data Mining]], [[Machine Learning]] || UI Developer & Software Engineering Student at [[Sam Houston State University]]
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|[[User:Yecril|Christopher Yeleighton]] ([[User_talk:Yecril|T]] [[Special:Contributions/Yecril|C]]) || [[Combinatorics]], [[Graph theory]] || Currently devoted to reviewing, fixing and typesetting
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|[[User:Mrseibert|Chad Seibert]] ([[User_talk:Mrseibert|T]] [[Special:Contributions/Mrseibert|C]]) || [[Number theory]], [[Group theory]], [[Abstract algebra]], [[Topology]], [[Linear Algebra]] || It's all on my user page
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|[[User:Clayt85|Clayt85]] ([[User_talk:Clayt85|T]] [[Special:Contributions/Clayt85|C]]) || Applied Mathematics, Mathematical Modeling of Physical and Biological Systems, Inverse Problems || Ph.D. Candidate in applied mathematics
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|[[User:Compsonheir|Compsonheir]] ([[User_talk:Compsonheir|T]] [[Special:Contributions/Compsonheir|C]]) || Computational fluid dynamics, free boundary problems, glacier mechanics || Pursuing a Ph.D. in applied math at the University of Washington
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|[[User:CRGreathouse|CRGreathouse]] ([[User_talk:CRGreathouse|T]] [[Special:Contributions/CRGreathouse|C]]) || Number theory, computational mathematics, applications to economics (esp. social choice theory/voting theory), cryptology, and computer science || Graduate student
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|[[User:Cronholm144|Cronholm144]] ([[User_talk:Cronholm144|T]] [[Special:Contributions/Cronholm144|C]]) || [[Calculus]], [[Linear algebra]], and [[Abstract algebra]] seem to be the types articles I have been writing lately. || Undergraduate student in mathematics at USC. After that, on to grad school!
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|[[User:C S|C S]] ([[User_talk:C S|T]] [[Special:Contributions/C S|C]]) || [[low-dimensional topology]] ||
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|[[User:CSTAR | CSTAR]] ([[User_talk:CSTAR|T]] [[Special:Contributions/CSTAR|C]])|| ||
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|[[User:Damelch|Damelch]] ([[User_talk:Damelch|T]] [[Special:Contributions/Damelch|C]]) || [[Combinatorics]], [[Graph Theory]], [[tournament (mathematics)|tournaments]] || Ph.D student at [[University of Colorado Denver]] with particular interest in [[tournament (mathematics)|tournaments]] and the [[Reconstruction conjecture]]
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|[[User:DRLB|Dan Brown]] ([[User_talk:DRLB|T]] [[Special:Contributions/DRLB|C]]) || [[Cryptography]], [[Combinatorics]], [[Algebra]], || PhD thesis was in Combinatorics. Now working in industry (cryptography).
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|[[User:Dan Hoey|Dan Hoey]] ([[User_talk:Dan Hoey|T]] [[Special:Contributions/Dan Hoey|C]]) || [[Combinatorial game theory]], [[Computer science]] applications, [[Computational mathematics]], [[Recreational mathematics]]||Working in CGT, wide mathematical interests.
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|[[User:Pooh4913 | Daniel Chang]] ([[User_talk:Pooh4913|T]] [[Special:Contributions/Pooh4913|C]])|| [[Multivariable calculus]], [[Linear Algebra]] || I just really like math.
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|[[User:Daniel C Sutton|Daniel C Sutton]] ([[User_talk:Daniel C Sutton|T]] [[Special:Contributions/Daniel C Sutton|C]]) || [[Group Theory]], [[Numerical Analysis]], [[Topology]], [[Linear Operators]],[[Fluid Mechanics]],[[Analysis of Real Valued Functions]] ||PhD Candidate, University of Bath
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|[[User:DavidCBryant|David Bryant]] ([[User_talk:DavidCBryant|T]] [[Special:Contributions/DavidCBryant|C]]) || Complex analysis, continued fractions, connections between analysis and algebra, the integers, and the history of mathematics. || Caltech graduate and former actuary with substantial assembly language programming experience.
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|[[User:David Eppstein|David Eppstein]] ([[User_talk:David Eppstein|T]] [[Special:Contributions/David Eppstein|C]]) || [[algorithms]], [[geometry]], [[graph theory]], [[combinatorics]], [[number theory]]||Comp. Sci. professor working in [[graph algorithms]] and [[computational geometry]]
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|[[User:Dr Marmilade|Dr Marmilade]] ([[User_talk:Dr Marmilade|T]] [[Special:Contributions/Dr Marmilade|C]]) || [[algorithms]], [[Chaos Theory]], [[Fractals]], [[Logarithms]]
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|[[User:Daviddaved|David Ingerman]] ([[User_talk:David Ingerman|T]] [[Special:Contributions/David Ingerman|C]]) || [[continued fractions]], [[graph theory]], [[inverse problems]]|| Great expanded mathematics articles in Wikipedia!
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|[[User:Dcterr|David Terr]] ([[User_talk:David Terr|T]] [[Special:Contributions/David Eppstein|C]]) || [[algorithms]], [[number theory]]||Math Ph.D. from UC Berkeley currently working as a [[software engineer]]
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|[[User:Dbenbenn | Dbenbenn]] ([[User_talk:Dbenbenn|T]] [[Special:Contributions/Dbenbenn|C]])|| [[geometric group theory]], [[knot theory]] || I'm happy to contribute [[MetaPost]] diagrams by request. I'm an administrator. Lately I've taken a break from working on math, though eventually I intend to add a lot about geometric group theory.
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|[[User:Dchmelik|dchmelik]] ([[User_talk:dchmelik|T]] [[Special:Contributions/dchmelik|C]].) ||everything, including [[history of mathematics]], [[philosophy of mathematics]], [[mathematical logic]] ||I am an undergraduate with skill in [[arithmetic]], [[geometry]], [[algebra]], [[mathematical analysis|analysis]], [[algorithms]], [[linear algebra]], [[discrete math]], [[fractal geometry]], [[dynamic systems]].
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|[[User:Dcoetzee | Dcoetzee]] ([[User_talk:Dcoetzee|T]] [[Special:Contributions/Dcoetzee|C]])|| [[abstract algebra]], [[graph theory]], [[topology]], [[real analysis]], [[complex analysis]], [[logic]], [[set theory]], [[number theory]] || I have a BS in math and have studied at a basic level abstract algebra, graph theory, topology, real and some complex analysis, logic, set theory, and number theory. If anyone needs help with pages in any of these areas please consult me.
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|[[User:Declan Davis|Declan Davis]] ([[User_talk:Declan Davis|T]] [[Special:Contributions/Declan Davis|C]])|| [[Differential geometry]], [[affine differential geometry]], [[singularity theory]], [[topology]] || Graduated with a PhD in 2008 in applications of singularity theory to affine differential geometry.
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|[[User:deeptrivia|deeptrivia]] ([[User_talk:deeptrivia|T]] [[Special:Contributions/deeptrivia|C]])|| [[numerical analysis]], [[Optimization (mathematics)|optimization]], [[number theory]], general stuff || PhD candidate in [[Mechanical engineering]]. I don't have a remarkable background in pure mathematics, but can contribute in areas of [[applied mathematics]], and work as a janitor.
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|[[User:Derek M|Derek M]] ([[User_talk:Derek M|T]] [[Special:Contributions/Derek M|C]]) || [[Computer Science]] ||
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|[[User:Designated Management| Designated Management]]||[[Fluid Dynamics]], [[Philosophy of Mathematics]], [[Real Analysis]], [[Topology]], [[Set Theory]]||BSc Mathematics and Philosophy & studying MSc Civil Engineering. Would like to work on expanding stubs & making articles more legible to a general audience (where this is possible & appropriate).
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|[[User:Dineshjk|Dinesh Karia]] ([[User_talk:Dineshjk|T]] [[Special:Contributions/Dineshjk|C]]) || [[Abstract Algebra]], [[Topology]], [[Real analysis]], [[Linear Algebra]], [[set theory]], [[Topological algebra]]s, [[Computer Programming]], [[Functional Analysis]], [[History of Mathematics]] || I am an [[India]]n. I did my Ph. D. in Topological Algebras. I have been working with the Department of Mathematics, [[Sardar Patel University]] for more than two decades. Presently, I have been working on e-content for the Students of Mathematics. I have authored a Textbook of Calculus.
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|[[User:Divespluto|Divespluto]] ([[User_talk:Divespluto|T]] [[Special:Contributions/Divespluto|C]]) || Education in, and history of mathematics. || Linguist etc. I've joined this group to help explain maths in general prose, and to learn more about maths myself.
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|[[User:Dmharvey|Dmharvey]] ([[User_talk:Dmharvey|T]] [[Special:Contributions/Dmharvey|C]]) || || I am an [[Australia]]n PhD candidate studying in the US. Serious interest is [[number theory]]. Minor interests include [[combinatorics]] and anything leaning towards [[computer science]]. Right now I'm working on a LaTeX => MathML converter ([[m:Blahtex|Blahtex]]) which may or may not one day be used in Wikipedia.
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|[[User:DocDeel516|DocDeel516]] --[[User:DocDeel516|DocDeel516]] [[User talk:DocDeel516|discuss]] 21:12, 23 December 2008 (UTC) ||[[Theoretical physics]], fractal geometry, algebra, mathematics || I am a 7th grader, taking eigth grade Algebra I. Planning on becoming a theoretical physics doctorate and professor. I have a hypotheses that I firmly believe might earn me that Ph.D. But if I must resort to a degree in mathematics, I have a hypotheses for that doctorate too.
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|[[User:Doctormatt|Doctormatt]] ([[User_talk:Doctormatt|T]] [[Special:Contributions/Doctormatt|C]]) || number theory, integer sequences, planar curves, graphics || I finished my PhD in 1997, in analytic number theory. Currently, I'm a mathematics lecturer.
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|[[User:DonkeyKong64|DonkeyKong64 (Mathematician in training)]] ([[User_talk:DonkeyKong64|T]] [[Special:Contributions/DonkeyKong64|C]]) || [[Group Theory]], [[Graph Theory]], [[Optimisation]], [[Optimal Control]], [[Stochastic Processes]]...and other stuff I can't think of right now || I'm doing honours in mathematics (as at 2007 - scheduled to finish honours in June 2008).
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|[[User:Donko XI|Donko XI]] ([[User_talk:Donko XI|T]] [[Special:Contributions/Donko XI|C]]) || [[Abstract Algebra|Algebra]] ||
|-
|[[User:Doug|Doug.]]([[User talk:Doug|talk]] • [[Special:Contributions/Doug|contribs]])||[[Number theory]], [[set theory]], [[geometry]], [[topology]], [[History of mathematics]]|| hobbiest, not formally trained in math except some basic statistics; I edit broadly
|-
|[[User:Download|download]] ([[User talk:Download|T]] [[Special:Contributions/Download|C]]) || [[Everything]] || Won numerous mathematics competitions
|-
|[[User:DrPhosphorus|DrPhosphorus]] ([[User talk:DrPhosphorus|T]] [[Special:Contributions/DrPhosphorus|C]]) || [[Nonlinear analysis, mathematical modeling]] || Math Professor
|-
|[[User:Dtaquinas|Dtaquinas]] ([[User_talk:Dtaquinas|T]] [[Special:Contributions/Dtaquinas|C]]) || [[Probability theory]], [[Random matrix|Random matrix theory]], [[Mathematical physics]], [[Integrable system|Integrable systems]] || Ph.D. student at the University of Arizona
|-
|[[User:Dylan Lake|Dylan Lake]] ([[User_talk:Dylan Lake|T]] [[Special:Contributions/Dylan Lake|C]]) || [[Mathematical analysis]], [[number theory]], [[cryptography]], [[Logic|logic (all kinds)]], [[History of mathematics]] ||
|-
|[[User:D.Lazard|D.Lazard]] ([[User_talk:D.Lazard|T]] [[Special:Contributions/D.Lazard|C]]) || [[Computer algebra]] and most areas of mathematics, with emphasis on effective and computational aspects|| Emeritus professor in mathematics and computer science
|-
|[[User:D.vegetali|D.vegetali]] ([[User_talk:D.vegetali|T]] [[Special:Contributions/D.vegetali|C]]) || [[Stochastic processes]], everything else || Math researcher
|-
|[[User:Ebony Jackson|Ebony Jackson]] ([[User_talk:Ebony Jackson|T]] [[Special:Contributions/Ebony Jackson|C]]) || Almost anything ||
|-
|[[User:Ecaterina Howard | Ecaterina Howard]] ([[User_talk:Ecaterina Howard|T]] [[Special:Contributions/Ecaterina Howard|C]]) || [[Computer science]], [[Topology]], [[Differential Geometry]], [[Information Theory]], [[Mathematical analysis and Calculus]], [[Philosophy of mathematics]], [[Computational mathematics]], [[Mathematical physics]], [[Logic]], [[History of mathematics]] || Theoretical Astrophysics research student at Macquarie University, Sydney.
|-
|[[User:editor0000001|editor0000001]] ([[User_talk:editor0000001|T |]] [[Special:Contributions/editor0000001|C]]) || Likes math.Joined because I think collaboration on math can lead to new things to learn. || I don't know what I know, but will add when I can.
|-
|[[User:EdJohnston|EdJohnston]] ([[User_talk:EdJohnston|T]] [[Special:Contributions/EdJohnston|C]]) ||[[Statistics]], [[Computer science]], [[Floating point arithmetic]], [[Logic]], [[History of mathematics]] ||
|-
|[[User:Eigenbra|Eigenbra]] ([[User_talk:Eigenbra|T]] [[Special:Contributions/Eigenbra|C]]) || [[Discrete geometry]], [[Mathematical physics]], [[Packing problems]] ||
|-
|[[User:ElectricRush|electricRush]] ([[User_talk:ElectricRush|T]] [[Special:Contributions/ElectricRush|C]]) (inactive)|| Math competitions, mathematics in general || In math levels 3 grades above my current grade level. Participates in various math competitions.
|-
|[[User:Elroch|Elroch]] ([[User_talk:Elroch|T]] [[Special:Contributions/Elroch|C]]) || [[Topology]], [[Geometry]], [[Differential geometry]], [[Real analysis]], [[Complex analysis]], [[Functional analysis]], [[Category theory]] || More of an ex-mathematician than a mathematician these days. Perhaps my deepest fascination is the mathematics that underlies our physical universe.
|-
|[[User:EmilJ|EmilJ]] ([[User_talk:EmilJ|T]] [[Special:Contributions/EmilJ|C]]) || [[mathematical logic]], mathematics in general || I'm a researcher at the Institute of Mathematics of the [[Academy of Sciences of the Czech Republic|Czech Academy of Sciences]].
|-
|[[User:Eric Herboso | Eric Herboso]] ([[User_talk:Eric Herboso|T]] [[Special:Contributions/Eric Herboso|C]])|| [[philosophy of mathematics]] || [[philosophy of mathematics]] and the like.
|-
|[[User:2^oscar|2^oscar]] ([[User_talk:2^oscar|T]] [[Special:Contributions/2^oscar|C]]) || Undergraduate mathematics, particularly pure. || Mathematics undergraduate. Unlikely to be able to contribute in any big way to more complex articles but will try to help out with small issues, such as typos, etc..
|-
|[[User:Evilbu|Evilbu]] ([[User_talk:Evilbu|T]] [[Special:Contributions/Evilbu|C]]) || (linear) algebra, number theory, geometry, history and geography || I am a Belgian student in his third year of pure mathematics. I still have to learn a lot but I like to contribute about the stuff I know. History and geography are my other interests but I am not a heavy contributor in that area I am afraid </nowiki>
|-
|[[User:Expz|Expz]] ([[User_talk:Expz|T]] [[Special:Contributions/Expz|C]]) || My specialities are [[Algebraic Geometry]], [[Algebra]], [[Geometry]], [[Differential Geometry]]. My interest is all of math, applied math and some theoretical physics. ||
|-
|[[User:Exteray|Exteray]] ([[User_talk:Exteray|T]] [[Special:Contributions/Exteray|C]]) || [[Geometry]], [[Probability]], [[Calculus]], [[Multivariable Calculus]], [[Statistics]], [[Discrete Mathematics]], [[Combinatorics]], [[Number Theory]] || High school student heavily involved in various mathematics competitions.
|-
|[[User:PikAsriel|PikAsriel]] ([[User_talk:PikAsriel|T]] [[Special:Contributions/PikAsriel|C]]) || Any Math || Math
|-
|colspan=3|[[#TC|1]] T = User's talk page, C = User's contributions.
|}

==Active participants F–J==

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
|-
|[[User:Fakedeeps|Fakedeeps]] ([[User_talk:Fakedeeps|T]] [[Special:Contributions/Fakedeeps|C]]) || [[History of mathematics|History]]; [[Philosophy of mathematics|Philosophy]]; [[Foundations of mathematics|Foundations]] || Looking forward to becoming an active Wikipedian. Generally all tasks acknowledged (though, currently short and easy ones preferred).
|-
|[[User:Fatka|Fatka]] ([[User_talk:Fatka|T]] [[Special:Contributions/Fatka|C]]) || [[Differential equations]], [[Linear algebra]] || I am graduate student in physics. I often edit articles that I find I can add to.
|-
|[[User:F4wk3s|Fawkes]] ([[User_talk:F4wk3s|T]] [[Special:Contributions/F4wk3s|C]]) || [[Algebraic Geometry]] || Graduate student interested in Algebraic Geometry.
|-
|[[User:Fieari|Fieari]] ([[User_talk:Fieari|T]] [[Special:Contributions/Fieari|C]]) || [[Discrete mathematics]], [[graph theory]], [[topology]], [[linear algebra]], [[Computer Science]] || It was suggested that I add myself to this list. I'm an undergraduate studying computer science, with an interest in some mathematical topics. As I'm currently taking a number of math courses, I've been using wikipedia as a study aid, and when I find that my textbook is more comprehensive than the article, I fix the article.
|-

|[[User:F=q(E+v^B)|F=q(E+v^B)]] ([[User_talk:F=q(E+v^B)|T]] [[Special:Contributions/F=q(E+v^B)|C]]) || [[Physics]], [[Applied maths]], [[Chaos theory]], all things [[Nonlinear system|non-linear]].
|| Undergraduate in [[mathematical physics]]. Shall refference, contribute to all within scope of knowledge, clean up every/anything.
|-

|[[User:Fractally|Fractally]] ([[User_talk:Fractally|T]] [[Special:Contributions/Fractally|C]]) ||[[Number theory]] [[Algebra]] [[Calculus]] [[Logic]]

|-

|[[User:Francesco Cattafi|Francesco Cattafi]] ([[User_talk:Francesco Cattafi|T]] [[Special:Contributions/Francesco Cattafi|C]]) || Mainly [[Differential geometry]] || I got my PhD from [[Utrecht University]] and I am currently a postdoctoral researcher
|-

|[[User:FrankP|FrankP]] ([[User_talk:FrankP|T]] [[Special:Contributions/FrankP|C]])
|| [[Mathematical logic|Logic]] [[Foundations of mathematics|Foundations]] [[abstract algebra|Algebra]]
|| Studied maths back when the world was young, now trying to remember if I knew anything
|-

|[[User:Fraqtive42|Fraqtive42]] ([[User_talk:Fraqtive42|T]] [[Special:Contributions/Fraqtive42|C]]) || [[Algebraic Topology]], [[Real Analysis]], [[Differential Equations]], [[Abstract Algebra]] (esp. [[Group Theory]]), [[Algebraic Number Theory]]|| I am an autodidact middle-schooler and math is my life. I am hoping to major in math. I am also interested in [[Philosophy of Politics]], [[Quantum Mechanics]], [[Astrobiology]], [[Organic Chemistry]], [[Criminology]], and [[Linguistics]].
|-
|[[User:Frayr|Frayr]] ([[User_talk:Frayr|T]] [[Special:Contributions/Frayr|C]]) || [[mathematical logic]]|| Undergraduate in philosophy with an interest in logic; listing myself here because some of the articles I'm editing fall within your purview.
|-
|[[User:Freenaulij|Freenaulij]] ([[User_talk:Freenaulij|T]] [[Special:Contributions/Freenaulij|C]])
|| Anything abstractral or hard to grasp. I enjoy doing math in my head. || High School Student who enjoys doing math competitions, learning about math, and highly enjoys math discussions.
|-
|[[User:FrozenJelloAndReason|FrozenJelloAndReason]] ([[User_talk:FrozenJelloAndReason|T]] [[Special:Contributions/FrozenJelloAndReason|C]])
|| In no particular order: Bayesian Statistics, Machine Learning, Numerical Analysis, Probability, Free Probability, Statistics, Computer Science, Functional Programming, Analysis, Functional Analysis, Random Matrix Theory
|| Math major who dabbles a lot in computer science and statistics
|-
|[[User:Footballrocks41237|Footballrocks41237]] ([[User_talk:Footballrocks41237|T]] [[Special:Contributions/Footballrocks41237|C]]) || [[Algebra]], [[Geometry]], [[Computer Science]], [[Trigonometry]] || Middle school student who does mathematics contests and competitions.
|-
|[[User:Fredrik|Fredrik]] ([[User_talk:Fredrik|T]] [[Special:Contributions/Fredrik|C]]) || Analysis, number theory, abstract algebra, computer algebra || Undergraduate student in [[engineering physics]].
|-
|[[User:FrederickII|FrederickII]] ([[User_talk:FrederickII|T]] [[Special:Contributions/FrederickII|C]]) || Geometry, Calculus, Graphing and [[Physics]] || Not a big fan of Math, but like exploring it.
|-
|[[User:Freiddie|Freiddie]] ([[User_talk:Freiddie|T]] [[Special:Contributions/Freiddie|C]]) || Anything within my capability. || An AS level student who is interested in [[math]] & [[physics]].
|-
|[[User:Freepopcornonfridays|Freepopcornonfridays]] || [[Combinatorics]], [[Graph theory]], [[Set theory]], and pretty much all other [[Discrete mathematics|Discrete topics]] || I am a high scholer with extensive contest history. I'll see what I can do to help.
|-
|[[User:Fropuff | Fropuff]] ([[User_talk:Fropuff|T]] [[Special:Contributions/Fropuff|C]])|| [[differential geometry and topology]], [[Lie groups]], [[category theory]], [[topology]], [[abstract algebra]], [[theoretical physics]]||
|-
|[[User:Ftbhrygvn | Ftbhrygvn]] ([[User_talk:Ftbhrygvn|T]] [[Special:Contributions/Ftbhfrygvn|C]])|| [[Algebra]]||A secondary student interested in Maths
|-
||[[User:Functor_salad|Functor salad]] || Algebra, functional analysis, category theory, mathematics in general|| Math student
|-
|[[User:FunnyMan3595 | FunnyMan3595]] ([[User_talk:FunnyMan3595|T]] [[Special:Contributions/FunnyMan3595|C]])|| [[abstract algebra]] || I'm a freshman majoring in mathematics, but I already have quite a few courses under my belt. My specialty is abstract algebra.
|-
|[[User:futurebird| futurebird]] ([[User_talk:futurebird|T]] [[Special:Contributions/futurebird|C]])|| I enjoy learning about [[graph theory]]. || I'm a grad student just starting a program in pure math.
|-
|[[User:GaborPete | Gabor Pete]] ([[User_talk:GaborPete|T]] [[Special:Contributions/GaborPete|C]])|| [[Probability theory]] and related fields: [[statistical mechanics]], [[geometric group theory]], [[combinatorics]], [[dynamical systems]], etc... || I'm an assistant prof in Toronto.
|-
|[[User:GabrielVelasquez | Gabriel Velasquez]] ([[User_talk:GabrielVelasquez|T]] [[Special:Contributions/GabrielVelasquez|C]])|| [[Climatology]], [[Astrophysics]] and [[Extra solar planet]]s || I watch for errors in new planet temperature models.
|-
|[[User:gala.martin | gala.martin]] ([[User_talk:gala.martin|T]] [[Special:Contributions/gala.martin|C]])|| [[Analysis]] and [[Probability theory]] || I work on stochastic PDEs.
|-
|[[User:Gamall Wednesday Ida|Gamall Wednesday Ida]] ([[User_talk:Gamall Wednesday Ida|T]] [[Special:Contributions/Gamall Wednesday Ida|C]]) || [[Theoretical computer science]], [[discrete mathematics]] || cf. user page.
|-
|[[User:Gantiganti | Gantiganti]] ([[User_talk:Gantiganti|T]] [[Special:Contributions/Gantiganti|C]])|| [[Calculus]], [[History of Mathematics]], and [[Linear Algebra]] || I'm a high school student taking math classes at a local university.
|-
|[[User:Garald | Garald]] ([[User_talk:Garald|T]] [[Special:Contributions/Garald|C]])|| [[Analysis]] and [[Probability theory]] || I work on number theory and group theory.
|-
|[[User:Gauge|Gauge]] ([[User_talk:Gauge|T]] [[Special:Contributions/Gauge|C]])
|| [[algebraic geometry]], [[algebraic topology]], [[category theory]], [[homotopy theory]], [[cohomology theory|cohomology theories]], [[mathematical logic]], [[computer science]]
|| Graduate student working on a PhD. Interested in the big complicated machines of modern homotopy theory and extraordinary cohomology theories, as well as their applications to algebraic geometry.
|-
|[[User:Gauge boson| Gauge boson]] ([[User_talk:Gauge Boson|T]] [[Special:Contributions/Gauge boson|C]]) || Algebra2 || Undergraduate at University of Kansas. I enjoy Algebra and most things that involve it.
|-
|[[User:General Eisenhower|General Eisenhower]] ([[User talk:General Eisenhower|T]] [[Special:Contributions/General Eisenhower|C]])||All math is my favorite.||Math teacher to 30 pupils. Name of school is not allowed to go public.
|-
|[[User:Genusfour|Genusfour]] ([[User talk:Genusfour|T]] [[Special:Contributions/Genusfour|C]])||[[Combinatorics]], [[Discrete geometry]],[[Algebraic topology]], [[Mathematical logic]], [[Modal logic]], [[Automata theory]] || undergrad.
|-
|[[User:Geometry guy|Geometry guy]] ([[User_talk:Geometry guy|T]] [[Special:Contributions/Geometry guy|C]]) || [[Differential geometry]], and anything related: [[algebraic geometry]], [[linear algebra]], [[differential calculus]] || I'm a professor of mathematics in the UK (but prefer to be judged on my edits, not my credentials!)
|-
|[[User:Gilderien|Gilderien]] ([[User_talk:Gilderien|T]] [[Special:Contributions/Gilderien|C]]) || [[Calculus]], [[Number Theory]] || I'm a student studying Mathematics at A-level in the UK
|-
|[[User:Gluons12|Gluons12]] ([[User_talk:Gluons12|T]] [[Special:Contributions/Gluons12|C]]) ||[[Number theory]] || Recreational mathematics
|-
|[[User:GManNickG|GManNickG]] ([[User_talk:GManNickG|T]] [[Special:Contributions/GManNickG|C]]) ||[[Mathematical logic]] || Minor in mathematics, autodidact in mathematical logic.
|-
|[[User:Godfrey 'Godfather' Tshehla|Godfrey 'Godfather' Tshehla]] ([[User_talk:Godfrey 'Godfather' Tshehla|T]] [[Special:Contributions/Godfrey 'Godfather' Tshehla|C]])||Pure and Applied Mathematics.|| If our lives cannot be described by all mathematics then we don't exist.I am studying Bsc in Computing and Mathematics at [[University of Witwatersrand]]
|-
|[[User:Googolplexideas|Manjil Saikia]] ([[User_talk:Googolplexideas|T]] [[Special:Contributions/Googolplexideas|C]])
|| [[number theory]], [[combinatorics]], [[discrete mathematics]]
|| Math student at [[ICTP]], has a masters in math from [[Tezpur University]].
|-
|[[User:GregWoodhouse|Greg Woodhouse]] ([[User_talk:GregWoodhouse|T]] [[Special:Contributions/GregWoodhouse|C]])
|| [[differential topology]], [[differential geometry]], [[number theory]], [[mathematical logic]], [[computer science]], [[mathematical physics]]. I realize that list is, well, a bit "eclectic"
||Though my education is in mathematics, I currently work in [[health informatics]] as a software developer.
|-
|[[User:Gro-Tsen|Gro-Tsen]] ([[User_talk:Gro-Tsen|T]] [[Special:Contributions/Gro-Tsen|C]])
|| [[algebraic geometry]] ([[rational variety|rational varieties]]), [[number theory]], [[axiomatic set theory]], [[metamathematics]] of [[Peano arithmetic|arithmetic]]
|| [http://tel.ccsd.cnrs.fr/documents/archives0/00/00/98/87/ PhD] from the [[University of Paris-Sud]] in 2005 (on the arithmetic of [[cubic hypersurface]]s), currently teaching at the [[École Normale Supérieure]] in Paris.
Main interest in [[algebraic geometry]], but very keen on [[mathematical logic]] as well.
|-
|[[User:Grubb257|Grubb257]] ||[[analysis]], [[physics]] || I'm a professor of mathematics and am completing a PhD in physics
|-
|[[User:Grubber|Grubber]] ([[User_talk:Grubber|T]] [[Special:Contributions/Grubber|C]]) || [[algebra]], [[stochastic process]], [[queueing theory]], [[electrical engineering]], [[steganography]] || I'm a graduate student in electrical engineering, studying stochastic processes, security, information theory, and algebra.
|-
|[[User:Guardian of Light|Guardian of Light]] ([[User_talk:Guardian of Light|T]] [[Special:Contributions/Guardian of Light|C]]) || [[number theory]], [[symmetric functions]], [[graph theory]], [[calculus]], [[differential equations]], [[algebraic topology]], [[algebra]], and [[category theory]] || I'm glad to be here helping as a part of the Mathematics project. My interests are largely in number theory and symmetric function theory, however my (professional level) expertise also extend to algebraic topology and category theory.
|-
|[[User:Correogsk|Gustavo Sandoval Kingwergs]] ([[User_talk:Correogsk|T]] [[Special:Contributions/Correogsk|C]])
|| [[history of mathematics]], [[mathematicians]]
|| BSc psychology ([[UNAM]], 1986-90). Master's degree in [[English (language)|Eng]]-[[Spanish (language)|Sp]]-Eng translation ([[Colmex]], 1990-92). Diploma courses: [[EEG]] ([[sleep medicine|sleep research]]), [[cognitive behavior therapy]], and [[terminology]]. Certified Eng-Sp-Eng [[translation|translator]] (Mexico City's Superior Court of Justice). In Wikipedia: Editing a) history/biographies of mathematics/-ians; b) difussion of maths (general public). Thanks and regards from [[Mexico City]]!
|-
|[[User:kjetil1001|Kjetil Halvorsen]] ([[User_talk:kjetil1001|T]] [[Special:Contributions/kjetil1001|C]])
|| [[mathemathical statistics]], [[probability]], [[linear algebra]], [[history of mathemathics]] || Master in mathemathical statistics, currently pursuing a PhD.
|-
|[[User:Hans_Adler|Hans Adler]] ([[User_talk:Hans_Adler|T]] [[Special:Contributions/Hans_Adler|C]])
|| [[model theory]], [[universal algebra]]
|| Postdoc in model theory. In Wikipedia: Working on the basic foundations of model theory and universal algebra.
|-
|[[User:Happysqirrel|Happy Squirrel]] ([[User_talk:Happysquirrel|T]] [[Special:Contributions/Happysqirrel|C]])
|| anything
|| Undergraduate student in pure math.
|-
|[[User:Harmonicmap|Harmonic Map]] ([[User_talk:Harmonicmap|T]] [[Special:Contributions/Harmonicmap|C]])
|| [[geometric topology]], [[algebraic topology]], [[differential geometry]], [[algebraic geometry]], [[symplectic geometry]]
||
|-
|[[User:Harsimaja|Harsimaja]] ([[User_talk:Harsimaja|T]] [[Special:Contributions/Harsimaja|C]])
|| [[geometric topology]], [[algebraic topology]], [[differential geometry]], [[algebraic geometry]], [[logic]], [[number theory]], [[group theory]], [[combinatorics]]
|| Doctoral student in geometric topology.
|-
|[[User:Helohe|Helohe]] ([[User_talk:Helohe|T]] [[Special:Contributions/Helohe|C]]) || [[category theory]], [[graph theory]], [[group theory]], [[topos theory]], [[computer science]] || I'm currently learning category and topos theory and can contribute from time to time. As soon I have read the [[Topos of Music]] I will extend the related articles.
|-
|[[User:Hennobrandsma|Henno Brandsma]] ([[User_talk:Hennobrandsma|T]] [[Special:Contributions/Hennobrandsma|C]]) || [[topology]] || I have a PhD in general topology, and have a braod interest in many branches of mathematics. I'll mostly focus on what I know best, i.e. topology. I might indulge my other passions, like [[linguistics]] and [[literature]].
|-
|[[User:Anish Mariathasan|Heptanitrocubane]] ([[User_talk:Anish Mariathasan|T |]] [[Special:Contributions/Anish Mariathasan|C]]) || Maths is key.
|-

|[[User:AndersHellstrom2|AndersHellstrom2]] ([[User_talk:AndersHellstrom2|T]] [[Special:Contributions/AndersHellstrom2|C]]) || ||
|-

|[[User:Hiabc|Hiabc]] ([[User_talk:Hiabc|T]] [[Special:Contributions/Hiabc|C]]) || [[abstract algebra]]; [[differential geometry]]; [[real analysis]]; [[algebraic topology]]; [[topology]] || At most 100% human.
|-
|[[User:s^2d^2|Hibbs, Max]] ([[User_talk:s^2d^2|T]] [[Special:Contributions/s^2d^2|C]]) || [[Algebra]],[[ Algebraic Topology]],[[ Geometric Topology]],[[ History of Mathematics]],[[ Mathematicians]],[[ Mathematics Education]] || Professor of Mathematics
|-
|[[User:Hirak_99|Hirak_99]] ([[User_talk:Hirak_99|T]] [[Special:Contributions/Hirak_99|C]]) || [[abstract algebra]],[[algebra]],[[calculus]],[[combinatorics]],[[discrete mathematics]],[[geometry]],[[measure theory]],[[probability]],[[set theory]],[[statistics]],[[topology]] || I have deep interest in almost all branches of Mathematics. I have done my BStat. and MStat. (i.e. Bachelors and Masters in [[Statistics]]) from [[Indian Statistical Institute]]. Would love to contribute in any place where I can.
|-
|[[User:Hv|hv]] ([[User_talk:Hv|T]] [[Special:Contributions/Hv|C]]) || [[number theory]], [[computation]], [[education]], [[OEIS|sequences]] || Primarily recreational since leaving university 20 years ago, I'm interested in all of maths but ignorant of most of it.
|-
|[[User:HyDeckar|HyDeckar]] || [[Statistics]], [[Econometrics]], [[Game Theory]], [[Mathematical finance]] || Honours Student in Statistics.
|-
|[[User:Hypergeometric2F1(a,b,c,x) | Hypergeometric2F1(a,b,c,x)]] ([[User_talk:Hypergeometric2F1(a,b,c,x)|T]] [[Special:Contributions/Hypergeometric2F1(a,b,c,x)|C]]) || [[Analytic number theory]], [[analysis]], [[number theory]], [[special functions]] || Math major at [[University of Oklahoma]], have since dropped out. I am currently an [[Autodidact]].
|-
|[[User:ianwmaggi|ianwmaggi]] ([[User_talk:ianwmaggi|T]] [[Special:Contributions/ianwmaggi|C]]) || [[Algebra 1]], || I'm a 9th Grade High School student that just loves mathematics and am trying to excel in math because I want to be a math teacher at high school when I'm older. Especially all forms of Calculus.
|-
|[[User:Igny|Igor]] ([[User_talk:Igny|T]] [[Special:Contributions/Igny|C]]) || [[differential equations]], [[dynamical systems]], [[numerical analysis]] || I have just joined the WP Math project.
|-
|[[User:Ikh|Ikh]] ([[User_talk:Ikh|T]] [[Special:Contributions/Ikh|C]]) || [[analysis]], [[differential equations|DEs]], [[theoretical physics]], [[control theory]]|| Math undergrad at [[University of Waterloo]]. Various other interests in math, but
|-
|[[User:IllQuill|IllQuill]] ([[User_talk:IllQuill|T]] [[Special:Contributions/IllQuill|C]]) || Mathematics || I am new to the project! Mathematics BS student
|-
| [[User:Ilovejames5|Ilovejames5]] ([[User_talk:Ilovejames5|T]] [[Special:Contributions/Ilovejames5|C]]) || numbers that explode your brain like [[googol]] || Joined wikipedia on 6 December 2022, i really like maths(math is my favourite subject), particularly large numbers like 13131313131313131.3131313131313
|-
|[[User:Indigopari|Indigopari]] ([[User_talk:Indigopari|T]] [[Special:Contributions/Indigopari|C]]) || All math! || I'm a math undergrad student currently taking Differential Equations, and I will be going into mathematics academia, but I haven't picked a particular field of math yet
|-
|[[User:IntegralPython|IntegralPython]] ([[User_talk:IntegralPython|T]] [[Special:Contributions/IntegralPython|C]]) || [[Quaternion]]s, [[Fractal]]s|| I have joined recently, and would appreciate any help I could get!
|-
|[[User:Iolaus221|Iolaus221]] ([[User_talk:Iolaus221|T]] [[Special:Contributions/Iolaus221|C]]) || [[Geometry]], [[Probability]], [[Algebra]], [[Algebraic Geometry]], [[Theoretical Physics]], [[Quantum Mechanics]], [[Neuroscience]] || Middle-school student involved with many different science and mathematics contests.
|-
|[[User:James3141592|James3141592]] ([[User_talk:James3141592|T]] [[Special:Contributions/James3141592|C]]) || ||
|-
|[[User:James in dc|James in dc]] ([[User_talk:James in dc|T]] [[Special:Contributions/James in dc|C]]) || [[Number theory]], some [[history of mathematics]]. || Mathematician major, now a retired programmer. Used to be active as Virginia-American.
|-
|[[User:Jordan Mitchell Barrett|Jordan Mitchell Barrett]] ([[User_talk:Jordan Mitchell Barrett|T]] [[Special:Contributions/Jordan Mitchell Barrett|C]]) || [[Mathematical logic]], [[category theory]], [[model theory]], [[Computability theory|computability]], [[combinatorics]], [[Ramsey theory]], [[Abstract algebra|algebra]]. || Mathematician (undergrad) at [[Victoria University of Wellington]]. Will provide minor edits and the occasional addition of a section / article if I see the need.
|-
|[[User:jflopezfernandez|Jose Fernando Lopez Fernandez]] ([[User_talk:jflopezfernandez|T]] [[Special:Contributions/jflopezfernandez|C]]) || [[Algebraic Topology]], [[Category Theory]], [[Epistemology]], [[Linguistics]] || Programmer
|-
|[[User:Jean Raimbault|jraimbau]] ([[User_talk:Jean Raimbault|T]] [[Special:Contributions/Jean Raimbault|C]]) || [[Geometric group theory]], [[differential geometry]], ... || I work at a math department in southern Europe, here I am mostly interested in research-level math pages concerning the areas I indicated and related topics
|-
|[[User: Jogesh 69|Jogesh 69]] ([[User_talk: Jogesh 69|T]] [[Special:Contributions/Jogesh 69|C]]) || [[Algebraic Topology]], [[Category Theory]], [[Epistemology]], [[Linguistics]] || Programmer
|-
|[[User:Sunyataivarupam|John]] ([[User_talk:Sunyataivarupam|T]] [[Special:Contributions/Sunyataivarupam|C]])|| [[philosophy of mathematics]], [[diagonal argument]]s, [[metalogic]], [[metamathematics]], [[set theory]], [[large cardinals]], [[foundations of mathematics]]|| Ph.D. in philosophy, math grad student. Researcher at a US public university.
|-
|[[User:Junesrose|Junesrose]] ([[User_talk:Junesrose|T]] [[Special:Contributions/Junesrose|C]]) || [[Vector calculus]], [[Mathematical physics]], [[Statistics]] || High school senior, have taken classes in single-variable calculus, applied mathematics, multi-variable calculus/vector calculus, geometry, and statistics. Hoping to go into aerospace engineering
|-
|colspan=3|[[#TC|1]] T = User's talk page, C = User's contributions.
|-
|}

==Active participants K–O==

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
|-
|[[User:K3bab | K3bab]] ([[User_talk:K3bab|T]] [[Special:Contributions/K3bab|C]])|| [[Numerical Analysis]], [[spectral methods]], [[fluid mechanics]]|| PhD student. Will try to contribute.
|-
|[[User:Kaboomkid01 | Kaboomkid01]] ([[User_talk:Kaboomkid01|T]] [[Special:Contributions/Kaboomkid01|C]])|| [[Geometry]], [[Algebra]] || I just feel like doing my job to help.
|-
| [[User:Kanogul|Kanogul]] ([[User talk:Kanogul|talk]]) || [[algebra]], [[geometry]], [[trigonometry]], [[calculus]], etc. || high school student in Pre-Calculus (as of December 2007), perfect score- math section [[PSAT]]
|-
| [[User:KathrynLybarger|KathrynLybarger]] ([[User talk:KathrynLybarger|T]] [[Special:Contributions/KathrynLybarger|C]]) || [[numerical linear algebra]], [[wavelet]]s || Librarian, [[Singular value decomposition|SVD]] enthusiast
|-
| [[User:Kaustuv|Kaustuv Chaudhuri]] ([[User talk:Kaustuv|T]] [[Special:Contributions/Kaustuv|C]]) || [[mathematical logic]], [[automated reasoning]], [[type theory]] || Math. logic postdoc
|-
|[[User:KennyDC|Kenny]] ([[User_talk:KennyDC|T]] [[Special:Contributions/KennyDC|C]]) || [[functional analysis]], [[operator algebras]], [[quantum groups]] || second year PhD
|-
|[[User:Ketsuekigata|Ketsuekigata]] ([[User_talk:Ketsuekigata|T]] [[Special:Contributions/Ketsuekigata|C]]) || [[calculus]], [[discrete mathematics]], [[wavelets]], [[cycloid|cycloids]], [[ballistics]], [[number theory]], [[asymmetric cryptography]], [[factorization]] || I'm just going into college, but I'll do what I can. Math is my answer to angst.
|-
|[[User:Kevin baas~enwiki | Kevin Baas]] ([[User_talk:Kevin_baas~enwiki|T]] [[Special:Contributions/Kevin_baas~enwiki|C]])|| [[fractional calculus]], [[Information geometry]] || I started the [[fractional calculus]] section. Though it is still embryonic, it is very much 'my style', which is still under development. -Also started [[Information geometry]] section. I am just learning about this, thought.
|-
|[[User:Kiefer.Wolfowitz | Kiefer.Wolfowitz]] ([[User_talk:Kiefer.Wolfowitz|T]] [[Special:Contributions/Kiefer.Wolfowitz|C]])|| [[Statistics]] and [[optimization theory]] & [[computational mathematics|computations]]; some [[mathematical economics|economics]] || [[Applied mathematics|Applications]] often involve [[pure mathematics]], e.g. [[operator theory]], [[probability theory|probability]] [[random vector|in]] [[Banach space]]s, [[oriented matroid|matroid]]s, etc.
|-
|[[User:Kier07|Kier07]] ([[User_talk:Kier07|T]] [[Special:Contributions/Kier07|C]])|| [[Galois theory]], [[algebraic topology]], [[Lie groups]] || I have always loved mathematics. I'm a senior undergraduate math major at Dartmouth College, soon to be a Ph.D. student in math at Northwestern University. I love the Wikipedia philosophy that (1) lots of information should be available in one place, (2) it should be accessible to the intelligent layperson, and (3) anyone can contribute. I want to be a part of it!
|-
|[[User:kilbad | kilbad]] ([[User_talk:kilbad|T]] [[Special:Contributions/kilbad|C]])|| [[calculus]]
|-
|[[User:King_Bee | King Bee]] ([[User_talk:King_Bee|T]] [[Special:Contributions/King_Bee|C]])|| [[graph theory]], [[relation algebra]]s, [[algebra]], [[logic]] || I'm a Ph.D. student at Iowa State University, and I want to contribute in a meaningful way to the mathematics articles here.
|-
|[[User:King_of_Hearts|King of Hearts]] ([[User_talk:King_of_Hearts|T]] [[Special:Contributions/King_of_Hearts|C]]) || [[number theory]], [[combinatorics]], [[real analysis]]|| Undergraduate math major at Stanford University.
|-
|[[User:Kmhkmh | Kmhkmh]]|| geometry, highschool math topics, education, math in general || I have a German degree in math (Diplom) and a master in education science. I basically contribute new articles wherever I see things missing, that i can fill in. Other than that occasional proof reading and factual corrections.
|-
|[[User:Kompik|Kompik]] ([[User_talk:Kompik|T]] [[Special:Contributions/Kompik|C]]) || [[category theory]], [[general topology]] || Ph.D. student, my work is concerned in categorial topology. Whenever I had enough time, I will work on the entries in wiki and planetmath as well, since I consider both very good projects.
|-
|[[User:krishnachandranvn|V N Krishnachandran]]  || [[Semigroup]]s, [[Geometry]], [[Topology]], [[Kerala school of astronomy and mathematics]] || Ph.D. from [[University of Kerala]], [[India]].
|-
| [[User:Kwerve| Kwerve]] || [[Number Theory]], [[Abstract Algebra]], [[Complex Analysis]] || Student at [[University of North Texas]].
|-
|[[User:L33tminion | L33tminion]] ([[User_talk:L33tminion|T]] [[Special:Contributions/L33tminion|C]])|| [[Computer engineering]], [[computer science]], [[formal systems]], [[Gödel's incompleteness theorem]], [[logic]] || currently studying [[Computer engineering]] at [[Olin College]].
|-
|[[User:Lantonov | Lantonov]] ([[User_talk:Lantonov|T]] [[Special:Contributions/Lantonov|C]])|| [[Combinatorics]], [[Coordinate systems]], [[Cosmology]], [[General relativity]], [[Mathematical physics]], [[Tensor algebra]], [[Vector algebra]] || M.Sc., Molecular Biology, Ph.D., Theoretical Physics from [[Virginia Tech]]
|-
|[[User:Chenxlee|Lee Butler]] ([[User_talk:Chenxlee|T]] [[Special:Contributions/Chenxlee|C]])||[[Number theory]], [[transcendence theory]]|| PhD student in the UK.
|-
|[[User:Laqy-peenu]] ([[User_talk: Laqy-peenu|T]] ||every thing in mathematics||-
|-
|[[User:Lcreight|Lee Creighton]] ([[User_talk:Lcreight|T]] [[Special:Contributions/Lcreight|C]])||[[statistics]], [[item response theory]], [[design of experiments]], general mathematics, mathematics teaching|| Ph.D., Mathematics Education, [[North Carolina State University]]
|-
| [[User:Leland_McInnes|Leland McInnes]] ([[User talk:Leland_McInnes|T]] [[Special:Contributions/Leland_McInnes|C]]) || [[algebra]], [[category theory]], [[cryptography]], [[logic]], [[topology]] || Ph.D. student working in algebra, particularly topological groups and (non-associative) rings.
|-
| [[User:LesMaxos|LesMaxos]] ([[User talk:LesMaxos|T]] [[Special:Contributions/LesMaxos|C]]) || [[abstract algebra]] and related areas || Graduate MMath student from the [[University of Edinburgh]].
|-
| [[User:Lesnail|LeSnail]] ([[User talk:Lesnail|T]] [[Special:Contributions/Lensnail|C]]) || Math! || Undergraduate math major at [[Oberlin College]]
|-
| [[User:Lester Mobley|Lester Mobley]] ([[User talk:Lester Mobley|T]] [[Special:Contributions/Lester Mobley|C]]) || ||
|-
|[[User:Lethe|lethe]] ([[User talk:Lethe|T]] [[Special:Contributions/Lethe|C]]) || ||
|-
|[[User:Liempt|Liempt]] ([[User_talk:Liempt|T]] [[Special:Contributions/Liempt|C]]) || All things math. || B. Sc. summa cum laude in cursu honorum, [[University of British Columbia]], M. Sc. summa cum laude (Thesis: Antinomies and formal logic) [[Oxford University]], Currently writing Ph. D. thesis at [[Oxford University]]
|-
|[[User:Linas | Linas]] ([[User_talk:Linas|T]] [[Special:Contributions/Linas|C]])|| [[quantum mechanics]], [[fractals]], [[modular forms]], [[Riemann surfaces]], [[number theory]], [[special functions]], [[quantum chaos]], [[zeta function]]s|| Trying to understand the nature of quantum [[wave function collapse]], which leads to the exploration of many remarkable and remarkably related topics in [[number theory]]. Strangely near the center of it all lies the [[theta function]] and the [[Hopf algebra]]s. Dabble in the [[Riemann hypothesis]] as well.
|-
|[[User:LittleDan | LittleDan]] ([[User_talk:LittleDan|T]] [[Special:Contributions/LittleDan|C]])|| [[geometry]], [[group theory]], [[vector spaces]] || I know up through [[geometry]], and a fair amount of [[group theory]] and [[vector spaces]]. I can usually pick things up from [[wikipedia]] articles, if not from [[mathworld]], then I can edit wiki articles for clarity.
|-
|[[User:Lord Shivan|Lord Shivan]] ([[User_talk:Lord Shivan|T]] [[Special:Contributions/Lord Shivan|C]]) || Anything to do with [[mathematics]] with all fields. || High school student.
|-
|[[User:LoveOfFate|LoveOfFate]] ([[User_talk:LoveOfFate|T]] [[Special:Contributions/LoveOfFate|C]]) || [[Probability]], [[Calculus]], [[Measure Theory]], [[Stochastic Calculus]],[[Differential Equations]], [[Numerical Linear Algebra]], [[Financial Mathematics]] ||MSc in Financial Engineering grad student.
|-
|[[User:Lucasstar1|Lucas Stonewell]] ([[User_talk:Lucasstar1|T]] [[Special:Contributions/Lucasstar1|C]]) || [[Calculus]] and [[Algebra]] || Have finished up to AP Calc BC, but struggle retaining information since I lack the time to practice daily.
|-
|[[User:Lucian_Chauvin|Lucian Chauvin]] ([[User_talk:Lucian_Chauvin|T]] [[Special:Contributions/Lucian_Chauvin|C]]) || [[Logic]], [[Theoretical Computer Science]], and many many more || Double major in pure mathematics and computer science at [[Texas A&M University]] (i love math :3)
|-
|[[User:Lunch|Lunch]]   ([[User_talk:Lunch|T]] [[Special:Contributions/Lunch|C]])|| [[numerical analysis]] and whatever catches my fancy ||
|-
|[[User:Machiavellian Eli|Machiavellian Eli]] ([[User_talk:Machiavellian Eli|T]] [[Special:Contributions/USERNAME|C]]) || [[Geometry]], [[Group theory]], [[Algebra]], [[Number theory]] and [[Multivariable calculus]] (though I'll happily talk about anything!) || Maths undergraduate at [[University of Oxford]]
|-
|[[User:Magidin | Magidin]]   ([[User_talk:Magidin|T]] [[Special:Contributions/Magidin|C]])|| [[group theory]], [[universal algebra]], [[number theory]] || I know some [[set theory]] and [[mathematical logic]] as well; [[number theory]] is more a hobby, my main work being in group theory and universal algebra.
|-
|[[User:Makotoy|Makotoy]] ([[User_talk:Makotoy|T]] [[Special:Contributions/Makotoy|C]]) || [[Operator algebra]] and related subjects ||
|-
|[[User:Googolplexideas|Manjil Saikia]] ([[User_talk:Googolplexideas|T]] [[Special:Contributions/Googolplexideas|C]])
|| [[number theory]], [[combinatorics]], [[discrete mathematics]]
|| Math student at [[ICTP]], has a masters in math from [[Tezpur University]].
|-
|[[User:Mathdoc|Mathdoc]] ([[User_talk:Mathdoc|T]] [[Special:Contributions/Mathdoc|C]]) || [[Algebra]], [[Statistics]], [[Number Theory]], [[Math Education]]|| [[CSUN ]]Mathematics Masters student.
|-
| [[User:MathMaven | MathMaven]] ([[User_talk:MathMaven|T]] [[Special:Contributions/MathMaven|C]]) ||[[Calculus]], [[Computer programming]], [[Computers]], [[Graphing calculator | Graphing calculators]], [[Logic]], [[Mathematics | Mathematics in general]], [[TI-Basic | TI-83+/84+ programming]] || I have studied mathematics up to the first year calculus level; I shall help wherever I am able to do so. I also love [[Wikipedia:WikiProject_Guild of Copy Editors | copy-editing]].
|-
|[[User:Marc Harper|Marc Harper]] ([[User_talk:Marc Harper|T]] [[Special:Contributions/Marc Harper|C]]) || [[algebra]], [[algebraic geometry]], [[algebraic topology]], [[category theory]]|| UIUC Mathematics PhD student.
|-
|[[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User_talk:Marc_van_Leeuwen|T]] [[Special:Contributions/Marc_van_Leeuwen|C]]) || [[combinatorics]], [[algebra]] || Member of the mathematics department in Poitiers (France).
|-
|[[User:Dominus | Mark Dominus]] ([[User_talk:Dominus|T]] [[Special:Contributions/Dominus|C]])|| ||
|-
|[[User:Mark viking | Mark viking]] ([[User_talk:Mark viking|T]] [[Special:Contributions/Mark viking|C]])|| [[Physics]], [[Applied Mathematics]], [[Mathematical programming]] || PhD in theoretical physics with an interest in mathematics and mathematical physics
|-
|[[User:Manabimasu| Manabimasu]] ([[User_talk:Manabimasu|T]] [[Special:Contributions/Manabimasu|C]])
||
||
|-
|[[User:MarkH21| MarkH21]] ([[User_talk:MarkH21|T]] [[Special:Contributions/MarkH21|C]])||[[algebraic geometry]], [[algebraic topology]], [[commutative algebra]], [[number theory]], [[representation theory]], [[history of mathematics]] || PhD in mathematics
|-
|[[User:Markus Krötzsch | Markus Krötzsch]] ([[User_talk:Markus Krötzsch|T]] [[Special:Contributions/Markus Krötzsch|C]])|| || I think many math articles still lack: general intros/motivation, links to relevant literature, objective account of alternative definitions (even if one definition is preferred in Wikipedia).
|-
|[[User:MarSch|MarSch]] ([[User_talk:MarSch|T]] [[Special:Contributions/MarSch|C]])
|[[geometry]], [[category theory]], [[physics]] || Working on my Master's thesis for physics and mathematics.
|-
|[[User:MartinDK|MartinDK]] ([[User_talk:MartinDK|T]] [[Special:Contributions/MartinDK|C]])
|| [[Applied Mathematics]], [[Mathematical programming]] and modelling, [[differential geometry]] || Masters degree in mathematical economics, specialized in finance.
|-
|[[User:Math Champion|Math Champion]] ([[User talk:Math Champion|T]] [[Special:Contributions/Math Champion|C]]) || Math competitions, such as [[MathCounts]] and [[American Mathematics Competitions|AMC]] (8,10, and 12). Also enjoy [[counting]], [[complex numbers]], [[logarithms]], [[graph theory]], and [[number theory]] || Winner of various state (Washington State Mathematics Championship) and national competitions (Rocket City Math League, and Math League); MathCounts State 12th place winner; AIME Participant,
|-
|[[User:Mathematici6n|Mathematici6n]] ([[User talk:Mathematici6n|T]] [[Special:Contributions/Mathematici6n|C]]) || || Currently a graduate student.
|-
|[[User:Mathemens|Mathemens]] ([[User_talk:Mathemens|T]] [[Special:Contributions/Mathemens|C]]) || Group theory, number theory, logics, statistics, chaos theory, applied math... many more. || Oh hai, I'm a self-taught math addict and an active WikiProject participant in future :).
|-
|[[User:MathMartin | MathMartin]] ([[User_talk:MathMartin|T]] [[Special:Contributions/MathMartin|C]])|| ||
|-
|[[User:Mathmo|Mathmo]] ([[User_talk:Mathmo|T]] [[Special:Contributions/Mathmo|C]]) || || My [[Mathmo|name]] says it all!
|-
|[[User:MathStatWoman|MathStatWoman]] ([[User_talk:MathStatWoman|T]] [[Special:Contributions/MathStatWoman|C]]) || [[probability theory]] and [[statistics]], especially [[empirical process]]es, [[order statistic]]s, Vapnik-Chervonenkis theory, [[measure theory]], [[distributions]], [[function spaces]], [[metrics]], [[infinity]], [[history of mathematics]], [[logic]] and [[foundations of mathematics]], [[computing]]|| I just joined the WikiProject Mathematics and I am still quite new to Wikipedia; see my user page to learn about me.
|-
|[[User:Matty j|Matthew Johnston]] ([[User_talk:Matty j|T]] [[Special:Contributions/Matty j|C]]) || [[projected differential equations]], [[variational inequality|variational inequalities]], [[chaos theory]], education || Current Master's student at the [[University of Guelph]], [[Canada]], starting my Ph.D. in the fall at the [[University of Waterloo]].
|-
|[[User:MaxineLund|MaxineLund]] ([[User_talk:MaxineLund|T]] [[Special:Contributions/MaxineLund|C]]) || [[set theory]], [[foundations of mathematics]], [[mathematical logic]], education || BSc. in mathematics, BA. in analytic philosophy; MA in philosophy focusing on mathematical logic and foundational problems of mathematics, ABD in philosophy, focusing on philosophy of mathematics, ontology of mathematical objects. personally interested in education of mathematics.
|-
|[[User:MC10|MC10]] ([[User talk:MC10|T]] [[Special:Contributions/MC10|C]]) || [[mathematics|general mathematics]], [[number theory]], groups, [[probability]], [[trigonometry]], [[algebra]] || Enjoy doing mathematics
|-
|[[User:Meekohi|Meekohi]] ([[User_talk:Meekohi|T]] [[Special:Contributions/Meekohi|C]]) || [[Graph theory]], [[cryptography]], and [[complex systems]] || I'll be trying to keep the collaboration of the week running if I can.
|-
|[[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User_talk:Meni Rosenfeld|T]] [[Special:Contributions/Meni Rosenfeld|C]]) || [[mathematical logic]], [[real analysis]] || BA degree in mathematics, interested especially in logic and the quest to make mathematical theories as precise and unambiguous as possible, but in virtually any other area of mathematics (and other sciences) as well.
|-
|[[User:Meineliebe97|Meineliebe97]] ([[User_talk:Meineliebe97|T]] [[Special:Contributions/Meineliebe97|C]]) || [[Associated legendre functions]], [[Transformations]], [[topology]], [[combinatorics]], [[probability integral]] and [[mathematical logic]], [[mathematical physics]] || Blogger,forum creator and passionate in Mathematics and Electrical Engineering
|-
|[[User:Mercrutio|Mercrutio]]([[User_talk:Mercrutio|T]]  [[Special:Contributions/Mercrutio|C]]) || [[Set Theory]], [[Topology]], [[Cryptography]] || Undergraduate Math Major new to Wikipedia. I'm loving Set Theory and Topology, and I'm interested in learning and contributing what I can.
|-
| [[User:Meshach|Meshach]] ([[User_talk:Meshach|T]] [[Special:Contributions/Meshach|C]]) || [[Logic]], [[Complex Numbers]], [[Computer Science]] || B.Sc. in Computer Science
|-
|[[User:Messagetolove|Messagetolove]] ([[User_talk:Messagetolove|T]] [[Special:Contributions/Messagetolove|C]])||[[abstract algebra]]||Professional Mathematician new to Wikipedia: (so far) trying to respect (modulo accuracy) and expand, existing edits.
|-
|[[User:Mets501|Mets501]] ([[User_talk:Mets501|T]] [[Special:Contributions/Mets501|C]]) || anything relating to math, particularly [[calculus]], [[computer science]], and [[elementary mathematics]] || I'm a high school student.
|-
|[[User:Mgnbar|Mgnbar]] ([[User_talk:Mgnbar|T]] [[Special:Contributions/Mgnbar|C]]) || [[geometry]], [[mathematical physics]] || Ph.D. in geometry
|-
|[[User:Michael Belisle| Michael Belisle]] ([[User_talk:Michael Belisle|T]] [[Special:Contributions/Michael Belisle|C]])|| [[Applied mathematics]], [[Fluid dynamics]], [[Nonlinear systems]], [[Stability theory]]||| I am a PhD student doing experimental research in laminar-turbulent transition.
|-
|[[User:bsodmike|Michael M. W. de Silva]] ([[User_talk:bsodmike|T]] [[Special:Contributions/bsodmike|C]])|| [[Nonlinear systems]], [[Numerical methods]], [[Numerical analysis]], [[Fundamental algebra]], [[Integral calculus]], [[Differential calculus]], [[Vector calculus]], [[Solid Geometry]], [[Trigonometry]], [[Physics]], [[Theoretical physics]], [[Parametric modelling]], [[Complex numbers]], [[Discrete mathematics]], [[Dynamical systems]], [[Linear algebra]], [[Advanced engineering mathematics]], [[Computer Science]], [[Computer engineering]], [[Electronic engineering]], [[Electrical engineering]], [[Mechatronics]]||| Qualifications: MSc (Dist) Mechatronics, BEng (Hons) Electronic and Computer Engineering ([http://www.mwdesilva.com resume])
|-
|[[User:Mihal Orela|Михал Орела]] ([[User_talk:Mihal Orela|T]] [[Special:Contributions/Mihal Orela|C]])||all math relating to [[computer science]]; special interest in [[category theory]] ||| formal qualifications (Europe): BSc Mathematics; PhD Computer Science.
|-
|[[User:Mhym| Mhym]] ([[User_talk:Mhym|T]] [[Special:Contributions/Mhym|C]])|| [[Geometry]] and [[Discrete Mathematics]]|| I am a Ph.D. student in the field.
|-
|[[User:Michael Hardy | Michael Hardy]] ([[User_talk:Michael Hardy|T]] [[Special:Contributions/Michael Hardy|C]])|| ||
|-
|[[User:mhender | Michael E. Henderson]] ([[User_talk:mhender|T]] [[Special:Contributions/mhender|C]])||[[Numerical Continuation]]||I work for IBM Research in the area of solving families of nonlinear equations, anf computational dynamical systems.
|-
|[[User:Miguel | Miguel]] ([[User_talk:Miguel|T]] [[Special:Contributions/Miguel|C]])|| ||
|-
|[[User:Mikkalai | Mikkalai]] ([[User_talk:Mikkalai|T]] [[Special:Contributions/Mikkalai|C]])|| [[graph theory]], [[computational geometry]], [[computer science]], [[operations research]], [[Optimization (mathematics)|optimization]], [[algorithm]]s|| I was trying to deal here with [[graph theory]], [[computational geometry]], [[computer science]], [[operations research]], [[Optimization (mathematics)|optimization]], [[algorithm]]s, which are areas of my expertise, but, I am being distracted by more obscure areas of knowledge, and math somehow slipped off my schedule.
|-
|[[User:Mindspillage | Mindspillage]] ([[User_talk:Mindspillage|T]] [[Special:Contributions/Mindspillage|C]])|| || [[Discrete math|Discretion]] is the better part of mathematics, at least for me. Just about done with my bachelor's.
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|[[User:Minirogue| Minirogue]] ([[User_talk:Minirogue|T]] [[Special:Contributions/Minirogue|C]])|| [[Mathematical beauty]], and anything interesting. || Working on my bachelor's in math, with plans to go to graduate school for math. I still haven't decided whether I want a master's or a doctorate, but I'll probably go for the doctorate as I probably won't be satisfied with the master's.
|-
|[[User:Mipchunk|Mipchunk]] ([[User_talk:Mipchunk|T]] [[Special:Contributions/Mipchunk|C]]) || Linear Algebra, Applied Mathematics, Differential Equations || I am an undergraduate student studying applied mathematics. As my studies progress my expertise will become more focused.
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|[[User:Mkortink|Mark Kortink]] ([[User_talk:Mkortink|T]] [[Special:Contributions/Mkortink|C]]) || Set Theory, Logic, Algebra, Imprecise Probability Theory || I am a maths tragic with an M.Sc in Maths and post grad DipORS (Operations Research and Statistics). My career has been in IT. I love Wikipedia, love Maths, want to contribute.
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|[[User:Mlliarm|Mlliarm]] ([[User_talk:Mlliarm|T]] [[Special:Contributions/Mlliarm|C]]) || [[Computational_mathematics|Computational mathematics]], [[Interval_analysis|Interval Analysis]], [[Logic]], [[Optimization]], [[Theoretical_computer_science|TCS]], [[Graph_theory|Graph theory]] || MSc in Computational Math. I'll help as much my time, knowledge and mood allow.
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|[[User:Model_Math|Model_Math]] ([[User_talk:Model_Math|T]] [[Special:Contributions/Model_Math|C]]) || Systems of Symbolic Equations, Knowledge Capture and Reasoning in Graphs, Formal Testing and Validation || I am old enough to remember hard-bound books and libraries but now thrilled by on-line knowledge; let's pass it all forward.
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|[[User:mppemberton|Mppemberton]] ([[User_talk:mppemberton|T]] [[Special:Contributions/mppemberton|C]])|| [[Commutative Algebra]], [[Algebraic Number Theory]], [[Algebraic Geometry]]||Current Ph.D. student in mathematics at University of Missouri-Columbia interested in commutative algebra and algebraic geometry.
|-
|[[User:Mrchapel0203|Mrchapel0203]] ([[User_talk:mppemberton|T]] [[Special:Contributions/mppemberton|C]])|| ||Current graduate student in mathematics at Hofstra University, working on thesis in mathematical modeling.
|-
|[[User:DrWikiWikiShuttle|DrWikiWikiShuttle]] ([[User_talk:DrWikiWikiShuttle|T]] [[Special:Contributions/DrWikiWikiShuttle|C]])|| optimization, algebraic combinatorics, polytopes, computer science, algorithms|| PhD in Applied Math from the University of California at Davis
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|[[User:MuZemike|MuZemike]] ([[User_talk:MuZemike|T]] [[Special:Contributions/MuZemike|C]])||[[Abstract algebra]], [[Number theory]]||[[Bachelor of Arts]] in mathematics from the [[University of Wisconsin in Madison]].
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|[[User:Mvitulli|Mvitulli]] ([[User_talk:Mvitulli|T]] [[Special:Contributions/Mvitulli|C]])||[[Commutative algebra]], [[Algebraic geometry]], women in mathematics|| Ph.D. in mathematics from the [[University of Pennsylvania]]
|-
|{{User:Myrecovery/Template:Myrecovery}}||'''[[Algebra]], [[Integral calculus]], [[Differintial calculus]], [[Vector graphics]], [[Solid Geometry]], [[Trigonometry]], [[Physics]].'''||'''Graduate in Atomic Physics and Mathematics from the Dhaka University.'''
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|[[User:Mythio|Mythio]] ([[User_talk:Mythio|T]] [[Special:Contributions/Mythio|C]])||[[Abstract Algebra]], [[Group Theory]], [[Cryptography]], [[Number Theory]]||Graduate student in Information Security Technology, consisting of a combination of [[Computer Science]] and [[Mathematics]].
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|[[User:Nat2|Nat2]] ([[User_talk:Nat2|T]] [[Special:Contributions/Nat2|C]]) || So far, basically everything I've come across. I'm most interested in [[Calculus]], [[Number theory]], and [[Cryptography]] with varying levels of knowledge in each. || I'm an undergraduate at the [[University of Chicago]], probably going to be a math major. Completed up to Calc III in high school but also learned outside material from math team, etc.
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|[[User:NatusRoma|NatusRoma]] ([[User_talk:NatusRoma|T]] [[Special:Contributions/NatusRoma|C]]) || [[graph theory]], [[group theory]], [[real analysis]], [[complex analysis]] || I am an undergraduate math major and aspiring mathematician.
|-
|[[User:Oldvlad55 |Vladimir Nazaikinskii]] ([[User_talk:Oldvlad55|T]] [[Special:Contributions/Oldvlad55|C]]) || [[mathematical physics]], [[semiclassical asymptotics]], [[noncommutative geometry]] || I am a Leading researcher at [[Ishlinsky Institute for Problems in Mechanics]], Moscow
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|[[User:NeilOnWiki|NeilOnWiki]] ([[User_talk:NeilOnWiki|T]] [[Special:Contributions/NeilOnWiki|C]]) || Whatever catches my eye. || Drawn to Pure Maths by nature; earnt a living from Applied.
|-
|[[User:nelsonalp| nelsonalp]] ([[User_talk:nelsonalp|T]] [[Special:Contributions/nelson|C]])|| [[probability]], [[algebra]], [[Statistics]]
|-
|[[User:Nerd1a4i|Nerd1a4i]] ([[User_talk:Nerd1a4i|T]] [[Special:Contributions/Nerd1a4i|C]]) || [[calculus]], [[linear algebra]], [[abstract algebra]], [[topology]], [[lie groups]] || I am a middle school student learning as much as I can in and out of school.
|-
|[[User:NerdyNSK|NerdyNSK]] ([[User_talk:NerdyNSK|T]] [[Special:Contributions/NerdyNSK|C]])|| [[Graph theory]] and [[discrete mathematics]].||BSc(Hons) Computer Science.
|-
|[[User:NereusAJ|NereusAJ]] ([[User_talk:NereusAJ|T]] [[Special:Contributions/NereusAJ|C]]) || [[Analysis]], [[Topology]], [[Number Theory]], [[Combinatorics]] || Undergraduate student in mathematics
|-
|[[User:Nevcamion|Nevcamion]] ([[User_talk:Nevcamion|T]] [[Special:Contributions/Nevcamion|C]])|| [[Differential Equations]], [[Mathematical Analysis]], [[Abstract Algebra]], and [[History of Mathematics]].||Undergraduate in Mathematics at Pitzer College.
|-
|[[User:Nightstallion|Nightstalllion]] ([[User_talk:Nightstallion|T]] [[Special:Contributions/Nightstallion|C]]) ||   || Four years of Austrian Mathematical Olympiads, will study Technical Mathematics starting October 2006.
|-
|[[User: Nishitpatira|Nishitpatira]] ([[Special:Contributions/Nishitpatira|C]]) || geometry, graph theory, game theory, algebra, analysis, probability, vectors ||
|-
|[[User:Nbarth|Nils Barth]] ([[User_talk:Nbarth|T]] [[Special:Contributions/Nbarth|C]]) || geometry, topology, algebraic geometry, elementary mathematics of all description, exposition ||
|-
|[[User:Nikolas Tales|Nikolas]] ([[User_talk:Nikolas Tales|T]] [[Special:Contributions/Nikolas Tales|C]])|| [[Mathematics education]] (and [[Science communication]]) ||
|-
|[[User:Nsk92|Nsk92]] ([[User_talk:Nsk92|T]] [[Special:Contributions/Nsk92|C]]) || [[geometric group theory]], [[geometric topology]]||
|-
|[[User:Nousernamesleft|Nousernamesleft]] ([[User_talk:Nousernamesleft|T]] [[Special:Contributions/Nousernamesleft|C]]) || [[calculus]], [[euclidean geometry]], [[algebraic geometry]] [[inequalities]] (in particular inequalities)|| High school freshman taking AP Statistics.
|-
|[[User:Ocolon|Ocolon]] ([[User_talk:Ocolon|T]] [[Special:Contributions/Ocolon|C]])|| [[applied mathematics]], [[number theory]], …|| Undergraduate student majoring in [[applied mathematics]].
|-
|[[User:Oleg Alexandrov | Oleg Alexandrov]] ([[User_talk:Oleg Alexandrov|T]] [[Special:Contributions/Oleg Alexandrov|C]])|| [[applied mathematics]]||
|-
|[[User:OllieThurston | Ollie Thurston]] ([[User_talk:OllieThurston|T]] [[Special:Contributions/OllieThurston|C]])|| Interested in whatever catches my eye || Currently still in high school, though I have completed the mathematics course, looking to study mathematics going forward.
|-
|[[User:Omerks|Omerks]] ([[User_talk:Omerks|T]] [[Special:Contributions/Omerks|C]]) || [[Algebra]], [[Algebraic number theory]] || I am a Ph.D. student at [[University of Massachusetts Amherst]] specializing in [[algebraic number theory]].
|-
|[[User:Oravec | Oravec]] ([[User_talk:Oravec|T]] [[Special:Contributions/Oravec|C]])|| [[computer science]], [[discrete mathematics]], [[T-theory]]|| I have a Bachelors in [[Computer Science]]. My current research area uses discrete mathematics, [[T-theory]] in particular.
|-
|[[User:Ozob|Ozob]] ([[User_talk:Ozob|T]] [[Special:Contributions/Ozob|C]])|||
|-
| colspan=3|[[#TC|1]] T = User's talk page, C = User's contributions.
|}

|-
|[[User:Memerman69|Memerman69]] ([[User_talk:memerman69|T]] 

==Active participants P–T==

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
|-
|[[User:Parrtech|Parrtech]] ([[User_talk:Parrtech|T]] [[Special:Contributions/Parrtech|C]]) || ||
|-
|[[User:ParticlePhysicsRules|ParticlePhysicsRules]] ([[User_talk:ParticlePhysicsRules|T]] [[Special:Contribution/ParticlePhysicsRules|C]])||[[Linear Algebra]], [[Topology]], [[Logic]]|| Currently working on my Masters Degree in Particle Physics
|-
|[[User:PatrickR2|PatrickR2]] ([[User_talk:PatrickR2|T]] [[Special:Contributions/PatrickR2|C]]) || [[general topology]], [[set theory]] || Ph.D. in mathematics
|-
|[[User:Paul August | Paul August]] ([[User_talk:Paul August|T]] [[Special:Contributions/Paul August|C]])|| [[topology]], [[category theory]], [[set theory]], [[logic]]|| After having gotten a Ph.D. in categorical topology in 1980, I immediately sold my soul to computers for 20 years. But Wikipedia, has now made me a Born Again mathematician ;-)
|-
|[[User:Pbroks13|Pbroks13]] ([[User_talk:Pbroks13|T]] [[Special:Contributions/Pbroks13|C]]) || Trigonometry, Logarithms, Calculus, Imaginary Numbers, Geometry || Will be majoring in mathematics starting next year.
|-
|[[User:peruvianllama | Peruvianllama]] ([[User_talk:peruvianllama|T]]  [[Special:Contributions/peruvianllama|C]]) || [[cellular automata]], [[computer science]], [[cryptography]], [[discrete mathematics]], [[history of mathematics]], [[number theory]] || I have an honours BSc degree in mathematics, where my thesis project dealt with linear cellular automata. I have some experience with programming, although mostly for non-math topics. For now, I'm mostly interested in contributing to math history articles.
|-
|[[User:PeterEasthope|PeterEasthope]] 
([[User_talk:PeterEasthope|T]] 
[[Special:Contributions/PeterEasthope|C]])
|| [http://members.shaw.ca/peasthope/#Problems Areas of interest]
|| [http://carnot.yi.org/ Comments]
|-
|[[User:PeterStJohn | Peter H. St.John]] ([[User_talk:PeterStJohn|T]]  [[Special:Contributions/PeterStJohn|C]]) || [[Enumeration]], [[number theory]], [[Genetic_algorithms|Gemetic Algorithms]], [[computer science]] || BA, MS, ton of additional grad school in math, but have mostly worked in software development. Right now I'm most interested in the GA for AI, and also nonlinear optimization, and Beowulfry to support the GA.
|-
|[[User:Phancy Physicist | Phancy Physicist]] ([[User_talk:Phancy Physicist|T]] [[Special:Contributions/Phancy Physicist|C]])|| All || As the name says, I am a physicist but a theoretical one so I know some mathematics as well.
|-
|[[User:Phenolla|Phenolla]] ([[User talk:Phenolla|talk]] · [[Special:Contributions/Phenolla|contribs]])
|[[Geometry]], [[number theory]]
|I'm majored in Computer Science and have interests in mathematics. I have basic understanding of Latex markup language which can used for editing mathematical formulae.
|-
|[[User:PicoMath | PicoMath]] ([[User_talk:PicoMath|Talk]] [[Special:Contributions/PicoMath|Contributions]]) || mainly [[Set Theory]], [[Abstract Algebra]], [[Real Analysis]] || Middle school student [[auto-didact|self-studing]] said subjects. Wikipedia editing does help me learn math, so I am sticking with it.
|-
|[[User:Pintoch | Pintoch]] ([[User_talk:Pintoch|T]] [[Special:Contributions/Pintoch|C]]) || [[Category theory]], [[computability theory]] || Graduate student in computer science.
|-
|[[User:Plaba123 | Plaba123]] ([[User_talk:Plaba123|Talk]] [[Special:Contributions/Plaba123|Contribs]])
||[[Abstract Algebra]], general math research.
||Current math student.
|-
|[[User:plw | Phil Wilson]] ([[User_talk:plw|T]] [[Special:Contributions/plw|C]])
|| [[applied mathematics]], [[mathematical modelling]], [[mathematical biology]], [[popular mathematics]], [[mathematics education]], [[philosophy of mathematics]], [[history of mathematics]]
|| For my background, interests, and pop math writing, see [http://www.math.canterbury.ac.nz/~p.wilson/]
|-
|[[User:PierreAbbat | Pierre Abbat]] ([[User_talk:PierreAbbat|T]] [[Special:Contributions/PierreAbbat|C]])|| || I've come up with a family of functions, called guyot functions, interpolating between the [[normal density function]] and the [[uniform density function]], which are pathological in that for many of them [[Taylor series]] don't work anywhere; a perfect [[spot function]] for halftones; and a conjecture about 64-100 sequences. I like to do mathematical experiments on the computer; hopefully I'll have more time for that...
|-
|[[User:Point-set topologist|Point-set topologist]] ([[User_talk:Point-set topologist|T]] [[Special:Contributions/Point-set topologist|C]]) || [[Point-set topology]], [[Set-theoretic topology]], [[Axiomatic set theory]], [[Algebraic topology]], [[Low-dimensional topology]], [[Noncommutative ring|Noncommutative ring theory]], [[Ring (mathematics)|Rings with involution]], [[Finite group theory]], [[Field theory]], [[Algebraic number theory]], [[Algebraic geometry]], [[Functional analysis]], [[Dynamical systems]], [[Category theory]], [[Probability theory]], [[Representation theory]], [[Finite mathematics]] and [[Fluid dynamics]]|| I am a [[mathematician]] who is interested in various branches of [[pure mathematics]] and [[theoretical physics]]
|-
|[[User:Polgoe|Polgoe]] ([[User_talk:Polgoe|T]] [[Special:Contributions/Polgoe|C]]) || [[Analytic number theory]], [[Combinatorial number theory]], [[Elementary number theory]]|| Working on analytic and elementary number theory, especially on arithmetic functions and diophantine equations
|-
|[[User:Potnisanish|Potnisanish]] ([[User_talk:Potnisanish|T]] [[Special:Contributions/Potnisanish|C]])
|| [[Scientific computing]], [[Engineering math]]|| I find a lot of the math pages to be extremely difficult to follow as introductions to subjects. I know wikipedia is meant to be a reference, but I think just a few Intuition sections scattered here and there can make a big difference in comprehensibility to technical laypeople who aren't mathematicians. (Engineering bachelors).
|-
|[[User:Pratyush Sarkar|Pratyush Sarkar]] ([[User_talk:Pratyush Sarkar|T]] [[Special:Contributions/Pratyush Sarkar|C]]) || [[Calculus]] || I am a high school student who is very interested in mathematics and physics. My goal is to become a physicist.
|-
|[[User:Project2501a|Project2501a]] ([[User_talk:Project2501a|T]] [[Special:Contributions/Project2501a|C]]) || [[Computer science]], [[Applied mathematics]], [[Chaos theory]], [[Information theory]], [[Linguistics]], [[Cryptography]], [[Discrete analysis]], [[Set theory]], [[Graph theory]], [[algorithm]]s, [[number theory]], [[Pattern Recognition]] || BS, Comptuter Science and Applied Mathematics from [[New Jersey Institute of Technology]], currently freelancing, trying to get back into the US :)
|-
|[[User:ProboscideaRubber15|ProboscideaRubber15]] ([[User_talk:ProboscideaRubber15|T]] [[Special:Contributions/ProboscideaRubber15|C]]) || [[Mathematical Biology]], [[Topology]], [[Logic]], [[History of Mathematics]], [[Computation]]|| I'm a math undergrad interested in most everything and hoping I know enough about something to make worthwhile contributions here.
|-
|[[User:Prokofiev2|Prokofiev2]] ([[User_talk:Prokofiev2|T]] [[Special:Contributions/Prokofiev2|C]]) ||[[Fractal geometry]], [[Chaos theory]]|| Independant french mathematician. Active since 2006.
|-
|[[User:Purpleleshi|Purpleleshi]] ([[User_talk:Purpleleshi|T]] [[Special:Contributions/Purpleleshi|C]]) || [[Chemistry]], [[Biomechanics]], [[mathematical physics]],[[Quantum Physics/mechanics]], [[set theory]], [[linear algebra]], [[Geometry]], [[calculus]], [[theoretical physics]], [[mathematical physics]], [[number theory]], . || I'm a senior in undergraduate college. I have taken all the possible math/physics/chemistry they offered here. Nothing excites me more anything, thats not because i am good at, infact i count myself as normal math level, but because nothing makes more sense than science and math for me.
|-
|[[User:Pustam.EGR|Pustam.EGR]] ([[User_talk:Pustam.EGR|T]] [[Special:Contributions/Pustam.EGR|C]]) ||All maths (esp. [[Engineering mathematics|Engineering mathematics]], [[Number theory|Number theory]], [[Combinatorics]], [[Probability and statistics|Probability and statistics]], [[Linear algebra|Linear algebra]], [[Calculus]], [[Geometry]], [[Mathematical analysis|Mathematical analysis]], [[Graph theory]], [[Chaos theory]], [[Numerical analysis|Numerical analysis]], [[Scientific computing]], [[Differential equation|Differential equations]], [[Recreational mathematics]], etc) || I love [[mathematics]] and [[physics]]. I believe mathematics is the foundation of all exact knowledge of [[List of natural phenomena|natural phenomena]] that effectively builds [[Problem solving|problem-solving skills]] and encourages [[logical thinking]] and mental [[rigour|rigour]].
|-
|[[User:Pyrop | Pyrop]] ([[User_talk:Pyrop|T]] [[Special:Contributions/Pyrop|C]])|| ||
|-
|[[User:Pyrospirit|Pyrospirit]] ([[User_talk:Pyrospirit|T]] [[Special:Contributions/Pyrospirit|C]]) || [[Axiom]]s, [[topology]], [[computer science]], [[set theory]], [[infinity]], [[linear algebra]], [[calculus]], [[theoretical physics]], [[mathematical physics]], [[number theory]], [[mathematical logic]], etc. || I'm a senior in high school taking undergraduate math. I will be majoring in math next year. I fully intend to continue doing mathematics my entire life.
|-
|{{User|Qxukhgiels}} || Mainly mathematics as it relates to physics, chemistry, and other branches of science, and also other misc. areas such as computer science, time, infinity, etc. || I was an advanced mathematics prep-school student, having self-educated myself and taken calculus at sixteen, along with several physics, chemistry, and biology classes earlier than usual. Never had lower than an A in a math class and a B in a science class (mostly A's).
|-
|[[User:Qwfp | Qwfp]] ([[User_talk:Qwfp|T]] [[Special:Contributions/Qwfp|C]]) || [[biostatistics]]||
|-
|[[User:Radlrb | Radlrb]] ([[User_talk:Radlrb|T]] [[Special:Contributions/Radlrb|C]]) || [[Philosophy of mathematics]], [[Logic]], [[Number theory]], [[Group theory]], [[Sequences]], [[Algebra|Algebra (associate and non-associative)]], [[Geometry|Eucledian and Hyperbolic geometry]] || Hello! I am here to contribute mainly on number articles, many of which can be largely expanded. I am working to publish a personal dissertation around 2025 based on novel properties of new and known [[mathematical object]]s that permit a greater understanding on the ''evolution'' of numbers from one onto another; as well as on central numbers that ''govern'' the expression and relationship between numbers (across mathematical realms), within specific and infinite domains.
|-
|[[User:RaitisMath| RaitisMath]] || ||
|-
|[[User:Ral315 | ral315]] ([[User_talk:Ral315|T]] [[Special:Contributions/Ral315|C]])|| || I'm an amateur mathematician, freshman in college, and creator of the [[Wikipedia:Wikiportal/Mathematics|Math WikiPortal]]. Check it out sometime!
|-
|[[User:RayAYang | RayAYang]] ([[User_talk:RayAYang|T]] [[Special:Contributions/RayAYang|C]])|| [[partial differential equations]], [[mathematical analysis]] || math graduate student
|-
|[[User:RayKiddy | RayKiddy]] ([[User_talk:RayKiddy|T]] [[Special:Contributions/RayKiddy|C]])|| || undergrad math, working in software.
|-
|[[User:Rcw258 | Rcw258]] ([[User_talk:Rcw258|T]] [[Special:Contributions/Rcw258|C]])|| [[arithmetic]], [[elementary geometry]] || Nothing, except I both love and am really good at math.
|-
|[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]])|| ||
|-
|[[User:Readro | Readro]] ([[User_talk:Readro|T]] [[Special:Contributions/Readro|C]])|| [[fluid dynamics]], [[partial differential equations]] || BSc (Hons) in mathematics, currently studying for an MSc in aerodynamics.
|-
|[[User:Redactedentity|Redactedentity]] || ||
|-
|[[User:Revolutionary girl euclid|Revolutionary girl euclid]] ([[User_talk:Revolutionary girl euclid|T]] [[Special:Contributions/Revolutionary girl euclid|C]]) || [[Pure Mathematics]] || College freshman with a broad interest in mathematics, especially pure math. New to editing Wikipedia.
|-
|[[User:Rey_grschel | Rey grschel]] ([[User_talk:Rey_grschel|T]] [[Special:Contributions/Rey_grschel|C]]) || [[calculus]], [[abstract algebra]], [[applied mathematics]], [[applied physics]], [[analytics]], [[theory of computation]] || High School Student, taking IB Mathematics HL
|-
|[[User:Reyk | Reyk]] ([[User_talk:Reyk|T]] [[Special:Contributions/Reyk|C]]) || a little bit of everything ||

|-
|[[User:Gill110951|Richard Gill]] ([[User talk:Gill110951|talk]])|| [[Statistics]], [[probability]], [[Life, the universe and everything]] || professor, university Leiden [http://www.math.leidenuniv.nl/~gill (my homepage)], infamous [[Richard D. Gill | Richard Gill]]
|-
|[[User:RJaguar3 | RJaguar3]] ([[User_talk:RJaguar3|T]] [[Special:Contributions/RJaguar3|C]])|| [[probability]] || Currently a senior in high school
|-
|[[User:RJBotting | RJBotting]] ( [[Special:Contributions/RJBotting|C]])|| [[formal logic]] [[discrete mathematics]] || B.Tech. Applicable Math and Ph.D. Computer Science
|-
|[[User:RLO1729 | RLO1729]] ([[User_talk:RLO1729|T]] [[Special:Contributions/RLO1729|C]])|| generalised [[number sequence|number sequences]], [[medical statistics]], [[optimal control theory]], [[Theory of relativity|special & general relativity]] || PhD optimal control, professor
|-
|[[User:RobHar|RobHar]] ([[User_talk:RobHar|T]] [[Special:Contributions/RobHar|C]]) || [[Number theory]], [[algebra]], [[mathematical physics]]|| Ph.D. student in [[algebraic number theory]]
|-
|[[User:RJE42 | Robin Evans]] ([[User_talk:RJE42|T]] [[Special:Contributions/RJE42|C]])|| [[statistics]], [[probability]], [[measure theory]] || Studying for the [[Part III of the Mathematical Tripos|CASM]] at [[University_of_Cambridge|Cambridge]], having graduated in maths from here last year.
|-
|[[User:RockMagnetist | RockMagnetist]] ([[User_talk:RockMagnetist|T]] [[Special:Contributions/RockMagnetist|C]])|| [[mathematical physics]] || I'm a geophysics professor interested in mathematics applied to geophysics and magnetism.
|-
|[[User:Rocky Mountain Goat | Rocky Mountain Goat]] ([[User_talk:Rocky Mountain Goat|T]] [[Special:Contributions/Rocky Mountain Goat|C]])|| [[mathematical analysis]] || B.A. in Mathematics
|-
|[[User:Root2 | Root2]] ([[User_talk:Root2|T]] [[Special:Contributions/Root2|C]])|| [[Euclidean Geometry]], [[Trigonometry]] ||
|-[[User:TheChard|Richard Thomas]] ([[User_talk:TheChard|T]] [[Special:Contributions/TheChard|C]]) || Set theory, logic, abstract algebra, field theory, Galois theory || Cambridge Mathematical Tripos graduate (2002 matric.)
|-
|[[User:RowanElder | RowanElder]] ([[User_talk: RowanElder|T]] [[Special:Contributions/RowanElder|C]])|| Statistics; differential equations; real and functional analysis; numerical simulation; category theory; applications to quantum and thermal physics || PhD Statistical Physics (theoretical, computational)
|-
|[[User:Russellvanderhorst|Russellvanderhorst]] ([[User talk:Russellvanderhorst|T]] [[Special:Contributions/Russellvanderhorst|C]])|| advanced [[applied mathematics]]||B.S. in applied mathematics
|-
|[[User:Ryan Reich|Ryan Reich]] ([[User_talk:Ryan Reich|T]] [[Special:Contributions/Ryan Reich|C]]) || [[Algebraic geometry]] || I'm a grad student at Harvard, interested in algebraic geometry. I tend to like sweeping overhauls, if I think I can pull them off competently.
|-
|[[User:Rybu|rybu]] ([[User_talk:Rybu|T]] [[Special:Contributions/Rybu|C]]) || [[geometric topology]] || I study manifolds, knots, links, spaces of embeddings, diffeomorphism groups and algorithms related to low-dimensional topology.
|-
|[[User:Pmanderson|Septentrionalis]] ([[User_talk:pmanderson|T]] [[Special:Contributions/Pmanderson|C]])
|| [[history of mathematics]], especially [[Greek mathematics]]; [[Abstract algebra]], [[number theory]]
| ABD mathemarics; straightened out [[Feynman_diagram#Mathematical_details]]; a little rusty on current work, but will try anything more like algebra than topology.
|-
|[[User:Sayantan m|Sayantan]] ([[User_talk:Sayantan m|T]] [[Special:Contributions/Sayantan m|C]])
|| [[Algebra]], [[Calculus]] etc.
| I'm just a mathematics student.
|-
|[[User:Salix alba|Salix alba]] ([[User_talk:Pfafrich|T]] [[Special:Contributions/Salix alba|C]]) né Pfafrich ([[Special:Contributions/Pfafrich|C]])|| [[Geometry]],[[Singularity theory]],[[Algebraic surface]]s,[[Polyhedron]],[[Multivariate statistics]],[[Procrustes analysis]],[[Computer vision]],[[Computer algebra system]]s, [[Scientific visualization]] || Real name Richard Morris, signing myself ''Salix alba''. A general theme of [[Shape]]: M.Sc, in [[Embedding|immersions]] of [[manifold]]s ([[topology]]), Ph.D. in [[Singularity theory]]/[[Scientific visualization]]. Post docs in [[Multivariate statistics]] and [[Computer vision]]. Written software for visualising singular algebraic surfaces and a Java equation parsing library.
|-
|[[User:Sardoodledom|Sardoodledom]] ([[User_talk:Sardoodledom|T]] [[Special:Contributions/Sardoodledom|C]]) || math competitions, random mathematical topics || I'm a high school math student and an active member of my school's math team.
|-
|[[User:Sceptre]] ([[User_talk:Sceptre|T]] [[Special:Contributions/Sceptre|C]]) || Pure mathematics || [[Master of Mathematics]] degree.
|-
|[[User:schmock|Schmock]] ([[User_talk:schmock|T]] [[Special:Contributions/schmock|C]]) || [[Probability theory]] and related fields ||
|-
|[[User:Schizoid|Schizoid]] ([[User_talk:schizoid|T]] [[Special:Contributions/schizoid|C]]) || [[algorithms]], [[Information geometry]], [[data clustering]] || interested in algorithms, [[computational geometry]] and data mining
|-
|[[User:SetaLyas|SetaLyas]] ([[User_talk:SetaLyas|T]] [[Special:Contributions/SetaLyas|C]]) || [[Group theory]] and related fields || In the last year of a masters at Warwick University
|-
|[[User:Shemitz|Shemitz]] ([[User_talk:Shemitz|T]] [[Special:Contributions/Shemitz|C]]) || [[Logic]], [[Discrete Mathematics]] || In the future, I will be pursuing graduate education.
|-
|[[User:shotwell|shotwell]] ([[User_talk:shotwell|T]] [[Special:Contributions/shotwell|C]]) || [[Spectral theory]], [[Automorphic forms]], [[Representation theory]],[[Number theory]] ||
|-
|[[User:Shreevatsa|Shreevatsa]] ([[User_talk:Shreevatsa|T]] [[Special:Contributions/Shreevatsa|C]]) || [[Computer Science]], [[combinatorics]], (elementary) [[number theory]] || Was undergrad in maths and CS, now grad student in something else.
|-
|[[User:Sigma69 |Sigma69]] ([[User_talk:Sigma69|T]] [[Special:Contributions/Sigma69 |C]]) || [[Computer science]], [[Theoretical computer science]], [[Theory of computation]], [[Computability theory]] || Studying Maths, along with Physics, Chemistry and Biology, at [[A-Level]].
|-
|[[User:Silly rabbit |Silly rabbit]] ([[User_talk:Silly rabbit|T]] [[Special:Contributions/Silly rabbit|C]]) || [[geometric analysis]], [[differential geometry]], [[partial differential equations]], [[Cartan's equivalence method|some aspects]] of [[Cartan connection|Cartan theory]], [[conformal geometry]] ||
|-
|[[User:Simplifix |Simplifix]] ([[User_talk:Simplifix|T]] [[Special:Contributions/Simplifix|C]]) || [[geometry]] interpreted widely || Math prof(essional) at a UK University.
|-
|[[User:Sjcjoosten |Sjcjoosten]] ([[User_talk:Sjcjoosten|T]] [[Special:Contributions/Sjcjoosten|C]]) || discrete optimization / combinatorics, theorem proving || Master student, starting as PhD in January in Netherlands.
|-
|[[User:skcpublic|skcpublic]] ([[User_talk:skcpublic|T]] [[Special:Contributions/skcpublic|C]]) || [[Statistics]], [[Game theory]] || My Math interests are application oriented; particularly the patterns that can be deduced from large data sets.
|-
|{{user|Skeptos}} || [[Numerical Analysis]], [[Partial Differential Equation]], [[Ordinary Differential Equation]], [[Discontinuous Dynamical Systems]] || Researcher in numerical analysis and applied mathematics.
|-
|{{user|Skipper1931}} || [[Algebra]], [[Statistics]], [[Computer Science]] || I love math and think it is very interesting.
|-
|[[User:Sławomir Biały |Sławomir Biały]] ([[User_talk:Sławomir Biały|T]] [[Special:Contributions/Sławomir Biały|C]]) || Analysis ||
|-
|[[User:Sodin|Sasha]] ([[User talk:Sodin|T]] [[Special:Contributions/Sodin|C]]) || ||
|-
|[[User:Some Old Man|Some Old Man]] ([[User_talk:Some Old Man|T]] [[Special:Contributions/Some Old Man|C]]) || || I am just Some Old Man who likes his cane, rocking chair, and rational thought.
|-
|[[User:SonyWii|SonyWii]] ([[User_talk:SonyWii|T]] [[Special:Contributions/SonyWii|C]]) || [[Algebra]], [[Calculus]], [[Number Theory]], [[Ring Theory]], [[Geometry]] || A high school student who does competitive math, and has interests in algebra, geometry, and abstract algebra, namely ring theory.
|-
|[[User:SpiralSource|SpiralSource]] ([[User_talk:SpiralSource|T]] [[Special:Contributions/SpiralSource|C]]) || Mathematical physics, set theory, topology, information theory, category-theory, and mathematical logic ||
|-
|[[User:Sr13|Sr13]] ([[User_talk:Sr13|T]] [[Special:Contributions/Sr13|C]]) || [[Number theory]], [[Theoretical physics]] || A middle school student that has interests in calculus, number theory, and GUT.
|-
|[[User:State Investigator Kurt Hartman|Kurt Hartman]] || [[Mechanics]], [[Geometry]], [[Calculus]] || University student who is studying mathematics, but wants to study aerospace engineering in masters, and work in that field.
|-
|[[User:Stca74|Stca74]] ([[User_talk:Stca74|T]] [[Special:Contributions/Stca74|C]]) || [[Algebraic geometry]], [[Differential geometry]], [[Algebra]], [[Number theory]], [[Probability]] || PhD in maths from Oxford, currently working in investment banking
|-
|[[User:Stca74|StellaAthena]] ([[User_talk:StellaAthena|T]] [[Special:Contributions/Stellaathena|C]]) || [[Theoretical Computer Science]], [[Graph Theory]], [[Combinatorics]], [[Number theory]], [[Mathematical Logic]] ||
|-
|[[User:Stemdude]] ([[User_talk:Stemdude|T]] [[Special:Contributions/Stemdude|C]]) || [[Computational Complexity]], [[Computer Science]], [[Number Theory]], [[Linear Algebra]], [[Complex Analysis]] || Hi, I'm very happy to join the Mathematics project.
|-
|[[User:SujinYH|SujinYH]] ([[User_talk:SujinYH|T]] [[Special:Contributions/SujinYH|C]]) || [[Algebra]], [[Number theory]], [[History of mathematics]], [[Mathematics education]], [[Mathematics as a language]] || I'm an undergrad in mathematics (and French), and my mathematical interests are varied and subject to change at any time. Even though mathematics education interests me, I don't plan on teaching.
|-[[User:Sulthan90|Sulthan]]([[User_talk:Sulthan90|T]] [[Special:Contributions/Sulthan90|C]])||[[Probablity]],[[Staistics]] and [[Econometrics]]|| Ph.D Scholar , Active in research and publication [http://www.iamsulthan.in My Homepage]
|-
|[[User:Supyovalk|Supyovalk]] ([[User_talk:Supyovalk|T]] [[Special:Contributions/Supyovalk|C]]) || [[Number theory]] || A highschool student , learns for 1st degree of computer science.
|-
|[[User:TakuyaMurata|TakuyaMurata]]([[User_talk:TakuyaMurata|T]] [[Special:Contributions/TakuyaMurata|C]])|| [[Noncommutative harmonic analysis]] || a Ph.D. student
|-
|[[User:syifaerr|syifaerr]]([[User_talk:Syifaerr|T]] [[Special:Contributions/syifaerr|C]])|| [[analysis]] || student
|-
|[[User:TauNeutrino|TauNeutrino]]([[User_talk:TauNeutrino|T]] [[Special:Contributions/TauNeutrino|C]])|| [[Calculus]], [[differential equations]], and [[abstract algebra]], among others ||Undergraduate studying physics and mathematics at [[Michigan State University]]
|-
|[[User:Tazerenix|Tazerenix]]([[User_talk:Tazerenix|T]] [[Special:Contributions/Tazerenix|C]])|| [[Geometry]] ([[Differential geometry]], [[algebraic geometry]], [[complex geometry]], [[symplectic geometry]], [[geometric analysis]], [[gauge theory]], [[mathematical physics]]) || PhD student studying special metrics in complex geometry and gauge theory at [[London School of Geometry and Number Theory]]
|-
|[[User:Tcnuk|Tcnuk]] ([[User_talk:Tcnuk|T]] [[Special:Contributions/Tcnuk|C]])|| [[Harmonic analysis]] || Working for a PhD at The [[University of Birmingham]]
|-
|[[User:TedPavlic|TedPavlic]] ([[User_talk:TedPavlic|T]] [[Special:Contributions/TedPavlic|C]])|| [[Calculus]], [[differential equations]], and [[abstract algebra]], among others ||[[Control Systems]] graduate student at [[The Ohio State University]]
|-
|[[User:TheAgentBrothers|TheAgentBrothers]] ([[User_talk:TheAgentBrothers|T]] [[Special:Contributions/TheAgentBrothers|C]] || [[Algebra]], [[Geometry]], [[Probability]], [[Square number]], [[Complex number]], [[Logarithms]], and [[Square root]] || Very interested in math since I was 9 years old.
|-
|[[User:The Anome | The Anome]] ([[User_talk:The Anome|T]] [[Special:Contributions/The Anome|C]])|| [[applied mathematics]], [[computer science]]|| I have some applied mathematics and computer science knowledge, so I can help out on physics and engineering-related maths
|-
|[[User:thehalfone| thehalfone]] ([[User_talk:thehalfone|T]] [[Special:Contributions/thehalfone|C]]) || [[Mathematical logic|Logic]], [[Model theory]], [[o-minimality]] and links to [[Real algebraic geometry| real geometry]] || PhD in Model Theory
|-
|[[User:TheFibonacciEffect|TheFibonacciEffect]] ([[User talk:TheFibonacciEffect|talk]]) || Analysis, Linear Algebra, Physics, Nummerical Methods ||
|-
|[[User:Thenub314|Thenub314]] ([[User_talk:Thenub314|T]] [[Special:Contributions/Thenub314|C]]) || [[Mathematical_analysis|Analysis]], [[Partial Differential Equation|PDEs]], [[Probability]] || I am a mathematician working in Edinburgh.
|-
|[[User:thepurplelefant|thepurplelefant]] ([[User_talk:thepurplelefant|T]] [[Special:Contributions/thepurplelefant|C]]) || All topics in math || I am interested in learning more about math and helping Wikipedia grow.
|-
|[[User:The Roc 1217|The Roc 1217]] ([[User_talk:The Roc 1217|T]] [[Special:Contributions/The Roc 1217|C]]) || [[geometry]], [[linear algebra]], [[calculus]], [[number theory]], [[polynomials]] || I'm a highschool math student. I got held back because I went to a private school. If I would have gone t oa public school I would have finished Calculus in 8th grade (four years ahead of where I am now). I love math.
|-
|[[User:TheQ Editor | TheQ Editor]] ([[User Talk:TheQ_Editor|T]] [[Special:Contributions/TheQ_Editor|C]])||[[paradoxes]], [[Number Theory]], [[Mental math|Math Tricks]], [[Graph Theory]] || Math is one of the things that I love. I want to help contribute to it.
|-
|[[User:The Scarlet Letter|The Scarlet Letter]] ([[User_talk:The Scarlet Letter|T]] [[Special:Contributions/The Scarlet Letter|C]]) || [[topology]], [[abstract algebra]], [[mathematical analysis]], [[functional analysis]], [[measure theory]], [[differential equations]], [[graph theory]], [[differential geometry]], among others || I'm a highschool math student.
|-
|[User:TheRadPixel|TheRadPixel]] ([[User_talk:TheRadPixel|T]] [[Special:Contributions/TheRadPixel|C]]) || Nothing really. || I like maths, that's it.
|-
|[[User:thudso|thudso]] ([[User_talk:thudso|T]] [[Special:Contributions/thudso|C]]) || [[Analysis]], [[Partial Differential Equations|PDEs]], [[Solid Mechanics]], and much, much more. || I'm an undergrad maths student, going on PhD student.
|-
|[[User:thuytnguyen48|Thuy]] ([[User_talk:thuytnguyen48|T]] [[Special:Contributions/thuytnguyen48|C]]) || Applied Mathematics||
|-
|[[User:Tiled|Tiled]] ([[User_talk:Tiled|T]] [[Special:Contributions/Tiled|C]]) || [[dynamical systems]] || grad student
|-
|[[User:TJhei07|TJhei07]] ([[User_talk:Tiled|T]] [[Special:Contributions/Tiled|C]]) || Interested in subject matters regarding [[Algebra]], [[Geometry]], [[Statistics]], and [[Trigonometry]] || Senior Student of a Science and Engineering Section
|-
|[[User:Toby Bartels | Toby Bartels]] ([[User_talk:Toby Bartels|T]] [[Special:Contributions/Toby Bartels|C]])|| ||
|-
|[[User:Tom-hundred-% | Tom hundred %]] ([[User_talk:Tom-hundred-%|T]] [[Special:Contributions/Tom-hundred-%|C]])|| [[Model Theory]], [[Topology]], [[Mathematical Physics]] || I'm just a generic geek
|-
|[[User:Tompw | Tompw]] ([[User_talk:Tompw|T]] [[Special:Contributions/Tompw|C]])|| [[Algebra]], [[Geometry ]], [[Topology]] || Maths graduate, studied mixture of pure and physics-related. Wide range of interests. Currently doing a lot of [[Wikipedia:WikiProject_Mathematics/Wikipedia_1.0|WP 1.0]] work
|-
|[[User:Tomo|Tomo]] ([[User_talk:Tomo|T]] [[Special:Contributions/Tomo|C]])|| [[discrete mathematics]], [[theoretical computer science]]||
|-
|[[User:Tomruen|Tom Ruen]]([[User_talk:Tomruen|T]] [[Special:Contributions/Tomruen|C]])|| [[Geometry]]||B.S. Computer Science & Mathematics
|-
|[[User:Tomtomn00|Tomtomn00]]([[User_talk:Tomtomn00|T]] [[Special:Contributions/Tomtomn00|C]])|| [[Algebra]], [[Mathematical Physics]], [[Equations]], [[Computer Science]], [[Calculus]], [[Gravity]]. I will prefer to do these, but will do all if needed.
|-
|[[User:TooMuchMath|TooMuchMath]] ([[User_talk:TooMuchMath|T]] [[Special:Contributions/TooMuchMath|C]]) || [[Abstract algebra]], [[Computational group theory]]|| Starting a PhD program in math at Cornell this fall (2006).
|-
|[[User:Topology Expert|Topology Expert]]] ([[User_talk:Topology Expert|T]] [[Special:Contributions/Topology Expert|C]]) || [[Topology]], [[Functional Analysis]], [[Measure Theory]], [[Group Theory]], [[Linear Algebra]], [[Real Analysis]], [[Differential Geometry]], [[Algebraic Topology]], [[Differential Topology]] ||
|-
|[[User:Tparameter|Tparameter]] ([[User_talk:Tparameter|T]] [[Special:Contributions/Tparameter|C]]) || [[differential equations]], [[numerical analysis]], [[complex analysis]], [[computer science]] || B.S. Applied Math & B.A. Computer Science.
|-
|[[User:Trovatore|Trovatore]] ([[User_talk:Trovatore|T]] [[Special:Contributions/Trovatore|C]]) || [[Set theory]], [[Descriptive set theory]], [[Borel equivalence relation]]s, foundational philosophy || Real name: Mike Oliver. UCLA PhD, 2003.
|-
|[[User:Truthfulcynic|Truthfulcynic]] ([[User_talk:Truthfulcynic|T]] [[Special:Contributions/Truthfulcynic|C]]) || [[Algebra]], [[Analytic Geometry]], [[Calculus]], [[Linear Algebra]] || High school student taking courses in Calculus, Linear Algebra, Combinatorics and Graph Theory, Computer Science, and Physics. Sysop at [http://www.proofwiki.org ProofWiki].
|-
|[[User:Tsirel|Tsirel]] ([[User_talk:Tsirel|T]] [[Special:Contributions/Tsirel|C]]) || [[Probability theory]] (mostly) || [[Boris Tsirelson]].
|-
|-
| colspan=3|[[#TC|1]] T = User's talk page, C = User's contributions.
|}

==Active participants U–Z==

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
|-
|[[User:UnbiasedBrigade|UnbiasedBrigade]] ([[User_talk:UnbiasedBrigade|T]] [[Special:Contributions/UnbiasedBrigade|C]]) || [[Computational linguistics]], [[Pure mathematics]] (although I'm not that advanced yet), [[Quantum mechanics]], [[Astrophysics]], [[Complex number]] theory (especially higher-dimensional analogs and other types of imaginary numbers), and probably other stuff I can't remember right now.|| Is currently in the second half of a calculus 1 course, and has done a bunch of looking into complex numbers and other number groups (hyperreals, quaternions, etc.). Trying to create a number system where div0 is allowed and non-contradictory.
|-
|[[User:Ushau97|Ushau97]] ([[User_talk:Ushau97|T]] [[Special:Contributions/Ushau97|C]]) || [[Mathematics]] (all) || Math student .
|-
|[[User:Vaughan Pratt|Vaughan Pratt]] ([[User_talk:Vaughan Pratt|T]] [[Special:Contributions/Vaughan Pratt|C]]) || [[Algebraic logic]], [[universal algebra]], [[category theory]], [[dynamic logic]], [[Chu space]]s, [[speech recognition]], [[natural language processing]], [[complexity theory]], [[computational geometry]], [[combinatorics]], [[quantum information]], [[thin film optics]] || Retired theoretical computer science prof.
|-
|[[User:VectorPosse|VectorPosse]] ([[User_talk:VectorPosse|T]] [[Special:Contributions/VectorPosse|C]]) || [[Contact geometry]], [[Symplectic topology]], [[Geometric topology]] || I am a professor of mathematics at [[Westminster College, Salt Lake City|Westminster College]] in [[Salt Lake City]], [[Utah]].
|-
|[[User:Viiticus|vitticus]] ([[User_talk:Viiticus|T]] [[Special:Contributions/Vitticus|C]]) || I specialize in [[Geometry]] and basic [[Algebraic]] subjects. || Math student interested in the [[ontological argument]] and its relation with Mathematics.
|-
|[[User:Christendom|Veritas]] ([[User talk:Christendom|talk]]) || Everything || I don't know alot about math; I'm a pathologist. However, I do admire it as the ideal pure science. Hopefully I can make some bangin' contributions.
|-
|[[User:Virginia-American|Virginia-American]] ([[User_talk:Virginia-American|T]] [[Special:Contributions/Virginia-American|C]]) || [[Number theory]] || Majored in math (no degree), program computers.
|-
|[[User:Vonkje| Vonkje]] ([[User_talk:Vonkje|T]] [[Special:Contributions/Vonkje|C]]) || [[computer science]], [[applied mathematics]], [[discrete mathematics]] || U.S.-born PhD student studying computer science in the Netherlands. Interested in modelling equilibrium-seeking and non-equilibrium seeking collective behaviors. Can be helpful in discrete mathematics. Mostly focuses on narrative precision.
|-
|[[User:Waltpohl | Walt Pohl]] ([[User_talk:Waltpohl|T]] [[Special:Contributions/Waltpohl|C]])|| || More years of graduate study in math than I really like to admit.
|-
|[[User:WalterMB | WalterMß]] ([[User_talk:WalterMB|T]] [[Special:Contributions/WalterMB|C]]) || [[Riemann surfaces]], [[group theory]], [[functional analysis]], [[algebraic topology]], [[harmonic analysis]], [[monodromy]], [[chaos theory]], [[fractals | Fractal]], [[Electromagnetics]] ||
|-
|[[User:Watcher of Forms | Watcher of Forms]] ([[User_talk:Watcher of Forms|T]] [[Special:Contributions/Watcher of Forms|C]]) || [[Algebra]], [[Theoretical Computer Science]] ||
|-
|[[User:Wcherowi|Wcherowi]] ([[User_talk:Wcherowi|T]] [[Special:Contributions/Wcherowi|C]]) || [[Finite geometry]], [[Projective Geometry]],[[Oval (projective plane)|Ovals and Arcs ]] ,[[Combinatorics]], [[History of mathematics]] || Professor of Mathematics, University of Colorado Denver
|-
|[[User:WDavis1911|WDavis1911]] ([[User_talk:WDavis1911|T]] [[Special:Contributions/WDavis1911|C]]) || [[Computer science]], [[Fuzzy set]] theory, [[Logic]] || Trained primarily in Computer science, but highly enmeshed in mathematics
|-
|[[User:Weston.pace|Weston Pace]] ([[User_talk:Weston.pace|T]] [[Special:Contributions/Weston.pace|C]]) || [[computer science]], [[mathematical analysis]], [[Linear Algebra]] || Math and Computer Science Student.
|-
|[[User:WillemienH|Willemien]] ([[User_talk:WillemienH|T]] [[Special:Contributions/WillemienH|C]]) || [[geometry]] || just like the subject.
|-
|[[User:William Ackerman|William Ackerman]] ([[User_talk:William Ackerman|T]] [[Special:Contributions/William Ackerman|C]]) || [[computer science]], [[special functions]], [[orthogonal polynomials]], [[mathematical analysis]] || Maths degree, but my professional field is computer science.
|-
|[[User:William2001|William]] ([[User_talk:William2001|T]] [[Special:Contributions/William2001|C]]) || -- || --
|-
|[[User:Wolfmankurd|wolfmankurd]] ([[User_talk:wolfmankurd|T]] [[Special:Contributions/wolfmankurd|C]]) || [[Pure mathematics]], [[proof]], [[mechanics]] || I'm only at AS-Level so I dont have very specific interests.
|-
|[[User:wvbailey|wvbailey]] ([[User_talk:wvbailey|T]] [[Special:Contributions/wvbailey|C]]) || Theory of and philosphy behind abstract computational machines/models and "[[algorithm]]", [[number theory]], logic, consciousness and computation, history of mathematics from 1850's, philosophy of mathematics and physics || B.A. Dartmouth 1970, B.Engg. Thayer School of Engineering (Dartmouth) 1971, MSEE Stanford (1975)
|-
|[[User:Xantharius|Xantharius]] ([[User_talk:Xantharius|T]] [[Special:Contributions/Xantharius|C]]) || [[Topology]], [[analysis]], [[geometry]], [[coarse geometry]], [[algebra]] || B.Sc. [[University of Edinburgh]] 1997, M.Sc. [[University of Tennessee]] 2006 (Thesis: "Coarse Structures and Higson Compactification"), currently in Ph.D. programme at [[Indiana University]]
|-
|[[User:XJamRastafire|xJaM]] ([[User_talk:XJamRastafire|T]] [[Special:Contributions/XJamRastafire|C]])
|| [[history of mathematics]], [[mathematical physics]], [[mathematician]]s, [[number theory]] ||
|-
|[[User:Yuanchosaan|Yuanchosaan]] ([[User_talk:Yuanchosaan|T]] [[Special:Contributions/Yuanchosaan|C]]) || Algebra, history of mathematics, quadratics, pi || I will attempt to help anyway I can, especially in the history area.
|-
|[[User:xDanielx|xDanielx]] ([[User_talk:xDanielx|T]] [[Special:Contributions/xDanielx|C]]) || [[addition]], [[subtraction]] || Might be able to help with [[multiplication]] if that's not too much of a stretch....
|-
|[[User:SpeedOfDarkness|SpeedOfDarkness]] ([[User_talk:SpeedOfDarkness|T]] [[Special:Contributions/SpeedOfDarkness|C]])
|| [[Mathematical analysis|Analysis]], [[Set theory]], [[Number theory]], [[Measure (mathematics)|Measure]], [[Fractal]], [[Mathematician|Mathematicians]] || 3rd year mathematics major and a wikinut :D
|-
|[[User:Yaris678|Yaris678]] ([[User_talk:Yaris678|T]] [[Special:Contributions/Yaris678|C]])
|| [[Applied mathematics]] || Undergrad degree was all sets and groups and analysis... but I much prefer the applied.
|-
|[[User:Xenomancer|Xenomancer]] ([[User_talk:Xenomancer|T]] [[Special:Contributions/Xenomancer|C]])
|| Life, the universe, and everything. || Engineer and scientist fluent in math with a tendency to wax poetic.
|-
|[[User:Yeetcode|Yeetcode]] ([[User_talk:Yeetcode|T]] [[Special:Contributions/Yeetcode|C]])
||[[Combinatorics]]
||Undergraduate student
|-
|[[User:Zfeinst|Zfeinst]] ([[User_talk:Zfeinst|T]] [[Special:Contributions/Zfeinst|C]])
|| [[Mathematical analysis|Analysis]], [[Applied mathematics]] || Undergrad in applied math, doctoral student in financial math
|-
|[[User:Zieglerk|Zieglerk]] ([[User_talk:Zieglerk|T]] [[Special:Contributions/Zieglerk|C]])
|| [[Symbolic computation]], [[Cryptography]], [[Game theory]] || Doctoral student in symoblic computation
|-
|[[User:Δθημτ π|Δθημτ π]] ([[User_talk:Δθημτ π|T]] [[Special:Contributions/Δθημτ π|C]]) || [[Geometry|geomertry]], [[Topology|topology]]. hate [[Number theory|numbers]]|| Ask about problemsolving and geomerty. topology is a interest, but dont ask me anything about it. '''dont ask about numbers'''.
|-
|-
| colspan=3|[[#TC|1]] T = User's talk page, C = User's contributions.
|}

==Former or inactive participants==
These users have either left the project or have not edited in the last three months. If you are listed here, please feel free to "reactivate" yourself on your return to active duty.

{| class="wikitable"
! User (T C)[[#note|1]] || Areas of interest || Comments
|-
|[[User:127|111111]] ([[User_talk:127|T]] [[Special:Contributions/127|C]]) ||   || I am the only member with a number for a name. :D
|-
|[[User:A_happybunny|A happybunny]] ([[User_talk:A_happybunny|T]] [[Special:Contributions/A_happybunny|C]]) || [[Mathematics]], [[Mathematics Education]], [[Euclid]] || I'm a third year undergrad studying BSc mathematics, mt main areas of interest are mathematics teaching and recently I have become quite interested by [[Eucild's Elements]]
|-
|[[User:...adam...|...adam...]] ([[User_talk:...adam...|T]] [[Special:Contributions/...adam...|C]]) ||[[Number Theory]], [[Coding Theory]], [[Cryptography]], [[Probability]]|| I am an undergrad reading maths in my final year at Cambridge University.
|-
|[[User:Adking80 | Adking80]] ([[User_talk:Adking80|T]] [[Special:Contributions/Adking80|C]])||[[graph theory]], [[combinatorics]], [[computational geometry]], [[computing theory]] || I'm a Ph.D. student, technically in computer science. You can expect lots of small edits from me, because I'm both pedantic and new at Wikipedia.
|-
|[[User:Aklippel | Aklippel]] ([[User_talk:Aklippel|T]] [[Special:Contributions/Aklippel|C]])||[[function theory]], [[number theory]], [[algebra]], [[functional analysis]] || I'm a mathematician from Germany, today working as senior engineer in computer networking. But still interested in mathematics and willing to improve Wikipedia
|-
|[[User:A. Catuneanu|A. Catuneanu]] ([[User_talk:A. Catuneanu|T]] [[Special:Contributions/A. Catuneanu|C]]) || [[Wave Theory]], [[Calculus]], [[Geometry]] || I am a student exploring mathematical relationships for the behaviour of waves in varying conditions and with varying properties. I'll contribute where I can!
|-
|[[User:Anabel Costa|Anabel Costa]] ([[User_talk:Anabel Costa|T]] [[Special:Contributions/Anabel Costa|C]]) || [[Algebra]], elementary mathematics || I am a Chemist and a Matheamtics teacher, I like to do some research in Algebra as a hobby.
|-
|[[User:Darkliight|Ben]] ([[User_talk:Darkliight|T]] [[Special:Contributions/Darkliight|C]]) || || Undergraduate Mathematics and Computer Science student. Will contribute where I can.
|-
|[[User:Bernard Hurley | Bernard Hurley]] ([[User_talk:Bernard Hurley|T]] [[Special:Contributions/Bernard Hurley|C]])|| [[algebra]], [[group theory]], [[algebraic topology]], [[mathematical logic]], [[foundations of mathematics]], [[philosophy of mathematics]], [[intuitionism]], [[Mathematical analysis|analysis]] || I was once a lecturer in mathematics but became disillusioned with the subject. After nearly thirty years my interest has been re-kindled.
|-
|[[User:Brozo M|Brozo M]] ([[User_talk:Brozo M|T]] [[Special:Contributions/Brozo M|C]]) ||[[mathematics]] || Senior student of Secondary school of electrical engineering and computing Maribor, Slovenia -> specializing in computing and informatics, programmer. I'm interested in making math web-applications.
|-
|[[User:Btg2290|btg2290]] ([[User_talk:Btg2290|T]] [[Special:Contributions/Btg2290|C]]) || Quadratics and Linear Algrebra || I'm in grade 11 university, so I haven't done very many things, we're mainly focusing on (I guess you could call it "simple") quadratics (such as <math>y = a^x</math>, <math>y = x^3</math> so yea, I'll contribute where I can :)
|-
|[[User:Bwyard|Bwyard]] ([[User_talk:Bwyard|T]] [[Special:Contributions/Bwyard|C]]) || ||
|-
|[[User:Robbjedi|Robb Carr]] ([[User_talk:Robbjedi|T]] [[Special:Contributions/Robbjedi|C]]) || || Student
|-
|[[User:Brian Chan|Brian Chan]] ([[User_talk:Brian Chan|T]] [[Special:Contributions/Brian Chan|C]]) || [[Probability Theory]], [[Geometric Brownian Motion]], [[Options]], [[Arbitrage pricing theory]], [[Black-Scholes Model]] || I'm currently an undergraduate studying Mathematical Finance.
|-
|[[User:Braindrain0000|Carl Peterson]] ([[User_talk:Braindrain0000|T]] [[Special:Contributions/Braindrain0000|C]]) || [[geometry]], [[set theory]], [[applied mathematics]] || Sophomore math major at [[Bellarmine University]]. Computer programmer by trade. Will take requests for geometry-based vector images.
|-
|[[User:Chancemill | Chance]] ([[User_talk:Chancemill|T]] [[Special:Contributions/Chancemill|C]])|| [[computer science]], [[formal system]]s, [[Gödel's incompleteness theorem]], [[computational complexity theory|complexity theory]], [[set theory]], [[number theory]] || I have a fair bit of knowledge in topics related to [[computer science]] and [[formal system]]s. I have keen learning interests in [[Gödel's incompleteness theorem]], [[computational complexity theory|complexity theory]], [[set theory]] and [[number theory]].
|-
|[[User:Chas_zzz_brown | Chas_zzz_brown]] ([[User_talk:Chas_zzz_brown |T]] [[Special:Contributions/Chas_zzz_brown |C]])|| [[group theory]], [[abstract algebra]] || My knowledge of topics outside of [[group theory]] is a [[monotonic|monotonically decreasing function]] of their relationship to [[abstract algebra]].
|-
|[[User:Chuchunezumi|Chuchunezumi]] ([[User_talk:Chuchunezumi|T]] [[Special:Contributions/Chuchunezumi|C]]) || My collegiate studies were primarily in continuous mathematics (analysis and topology), though I am also interested in some number theory, advanced linear algebra, and other algebraic topics. Furthermore, I have a decent background in modeling, focusing on structural geology. || While most of my editing contributions tend to be in other areas, I will be checking the list and evaluating good article candidates in mathematics.
|-
|[[User:Choni | Choni]] ([[User_talk:Choni|T]] [[Special:Contributions/Choni|C]])|| [[set theory]], [[logic]]|| Most recently, been working with a few set theory/logic articles. But lots of other stuff interests me as well...
|-
|[[User:Schopenhauer | Chopinhauer]] ([[User_talk:Schopenhauer|T]] [[Special:Contributions/Schopenhauer|C]])|| [[algebraic geometry]], [[number theory]] || Working in [[algebraic geometry]], [[number theory]], sometimes called [[arithmetic geometry]].
|-
|[[User:Mathmoclaire|Claire]] ([[User_talk: Mathmoclaire |T]] [[Special:Contributions/Mathmoclaire |C]]) || [[Dynamical systems]], [[bifurcation theory]], pattern formation. || I have a PhD from [[Cambridge University|Cambridge]] and am currently a postdoc at [[Northwestern University|Northwestern]].
|-
|[[User:Covington|Covington]] ([[User_talk:Covington|T]] [[Special:Contributions/Covington|C]])|| || Focusing on making mathematics articles understandable to most people.
|-
|[[User:CryptoDerk | CryptoDerk]] ([[User_talk:CryptoDerk|T]] [[Special:Contributions/CryptoDerk|C]])|| [[cryptography]], [[number theory]], [[coding theory]], [[modular arithmetic]], [[Galois field]]s, [[elliptic curve]]s || My research area is [[cryptography]], so I do a lot of work in [[number theory]], [[coding theory]], and stuff related to that, specifically modular arithmetic, Galois fields, and elliptic curves.
|-
|[[User:Cthulhu.mythos|Cthulhu.mythos]] ([[User_talk:Cthulhu.mythos|T]] [[Special:Contributions/Cthulhu.mythos|C]]) || [[Low-dimensional topology]] ||
|-
|[[User:Dan Gardner | Dan Gardner]] ([[User_talk:Dan Gardner|T]] [[Special:Contributions/Dan Gardner|C]])|| ||
|-
|[[User:Alodyne|Dave Rosoff]] ([[User_talk:Alodyne|T]] [[Special:Contributions/Alodyne|C]]) || [[algebraic topology]], [[homotopy theory]], [[triangulated category|triangulated categories]], [[algebraic number theory]], [[algebraic K-theory]], [[history of mathematics]] || I'm a Ph.D. student in mathematics. I mostly like to work on the topology and number theory articles. Please feel free to comment on my work in progress, at [[User:Alodyne|my user page]].
|-
|[[User:DavidDumas|David Dumas]] ([[User_talk:DavidDumas|T]] [[Special:Contributions/DavidDumas|C]]) || [[differential geometry]], [[complex analysis]]||
|-
|[[User:Decrypt3 | Decrypt3]] ([[User_talk:Decrypt3|T]] [[Special:Contributions/Decrypt3|C]])|| ||
|-
|[[User:DigitalCharacter | DigitalCharacter]] ([[User_talk:DigitalCharacter|T]] [[Special:Contributions/DigitalCharacter|C]])|| ||
|-
|[[User:DBJC|Dominic]] ([[User_talk:DBJC|T]] [[Special:Contributions/DBJC|C]]) || Most things you care to mention, but especially anything involving complex numbers || Currently reading Physics and Philosophy at undergraduate level at Oxford University
|-
|[[User:Dimension10|Dimension10]] ([[User_talk:Dimension10|T]] [[Special:Contributions/Dimension10|C]])<div style="font-size:70%">Indefinitely blocked</div> || [[Abstract algebra]], [[Clifford algebra]], [[Lie algebra]], [[Tensor algebra]], [[Vector calculus]]|| Some random person.
|-
|[[User:donludwig|Don Ludwig]] ([[User_talk:donludwig|T]] [[Special:Contributions/donludwig|C]]) || Ordinary differential equations, Partial differential equations, Mathematical Biology
||I am a retired Professor of Mathematics and Zoology
|-
|[[User:E=mc²|E=mc²]] ([[User_talk:E=mc²|T]] [[Special:Contributions/E=mc²|C]]) || [[Pure Mathematics]], [[Mathematical Logic]], [[Number Theory]], [[Cryptography]], [[Coding Theory]] ||
|-
|[[User:EulerGamma | EulerGamma]] ([[User_talk:EulerGamma|T]] [[Special:Contributions/EulerGamma|C]])
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Generating function

2024-01-06T21:35:17Z

Yeetcode: Expanded the Lagrange Inversion section. More yet to come.

{{Short description|Formal power series; coefficients encode information about a sequence indexed by natural numbers}}
{{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}
{{Very long|date=July 2022}}

In [[mathematics]], a '''generating function''' is a representation of an [[infinite sequence]] of numbers as the [[coefficient]]s of a [[formal power series]]. Unlike an ordinary series, the ''formal'' [[power series]] is not required to [[Convergent series|converge]]: in fact, the generating function is not actually regarded as a [[Function (mathematics)|function]], and the "variable" remains an [[Indeterminate (variable)|indeterminate]]. Generating functions were first introduced by [[Abraham de Moivre]] in 1730, in order to solve the general linear recurrence problem.<ref>{{cite book |author-link=Donald Knuth |first=Donald E. |last=Knuth |series=[[The Art of Computer Programming]] |volume=1 |title=Fundamental Algorithms |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |year=1997 |chapter=§1.2.9 Generating Functions}}</ref> One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

There are various types of generating functions, including '''ordinary generating functions''', '''exponential generating functions''', '''Lambert series''', '''Bell series''', and '''Dirichlet series'''; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in [[Closed-form expression|closed form]] (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate {{mvar|x}} may involve arithmetic operations, differentiation with respect to {{mvar|x}} and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of {{mvar|x}}. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of {{mvar|x}}, and which has the formal series as its [[series expansion]]; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a [[convergent series]] when a nonzero numeric value is substituted for {{mvar|x}}. Also, not all expressions that are meaningful as functions of {{mvar|x}} are meaningful as expressions designating formal series; for example, negative and fractional powers of {{mvar|x}} are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]. Generating functions are sometimes called '''generating series''',<ref>This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", ''[[Canadian Journal of Mathematics]]'' 3, [https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407 p. 405–411], but its use is rare before the year 2000; since then it appears to be increasing.</ref> in that a series of terms can be said to be the generator of its sequence of term coefficients.

==Definitions==

{{block quote
| text = ''A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.''
| author = [[George Pólya]]
| source = ''[[Mathematics and plausible reasoning]]'' (1954) }}

{{block quote
| text = ''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
| author = [[Herbert Wilf]]
| source = ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)}}

===Ordinary generating function (OGF)===

The ''ordinary generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">G(a_n;x)=\sum_{n=0}^\infty a_n x^n.</math>

When the term ''generating function'' is used without qualification, it is usually taken to mean an ordinary generating function.

If {{math|''a''''n''}} is the [[probability mass function]] of a [[discrete random variable]], then its ordinary generating function is called a [[probability-generating function]].

The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array {{math|''a''''m'',''n''}} (where {{mvar|n}} and {{mvar|m}} are natural numbers) is

<math display="block">G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n} x^m y^n.</math>

===Exponential generating function (EGF)===

The ''exponential generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.</math>

Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{harvnb|Flajolet|Sedgewick|2009|p=95}}</ref>

Another benefit of exponential generating functions is that they are useful in transferring linear [[recurrence relations]] to the realm of [[differential equations]]. For example, take the [[Fibonacci sequence]] {{math|{''fn''}<nowiki/>}} that satisfies the linear recurrence relation {{math|''f''''n''+2 {{=}} ''f''''n''+1 + ''f''''n''}}. The corresponding exponential generating function has the form

<math display="block">\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n</math>

and its derivatives can readily be shown to satisfy the differential equation {{math|EF{{pprime}}(''x'') {{=}} EF{{prime}}(''x'') + EF(''x'')}} as a direct analogue with the recurrence relation above. In this view, the factorial term {{math|''n''!}} is merely a counter-term to normalise the derivative operator acting on {{math|''x''''n''}}.

===Poisson generating function===
The ''Poisson generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{PG}(a_n;x)=\sum _{n=0}^\infty a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).</math>

===Lambert series===
{{main article|Lambert series}}
The ''Lambert series'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.</math>

The Lambert series coefficients in the power series expansions

<math display="block">b_n := [x^n] \operatorname{LG}(a_n;x)</math>

for integers {{math|''n'' ≥ 1}} are related by the [[Divisor sum identities|divisor sum]]

<math display="block">b_n = \sum_{d|n} a_d.</math>

The main article provides several more classical, or at least well-known examples related to special [[arithmetic functions]] in [[number theory]].

In a Lambert series the index {{mvar|n}} starts at 1, not at 0, as the first term would otherwise be undefined.

===Bell series===

The [[Bell series]] of a sequence {{math|''a''''n''}} is an expression in terms of both an indeterminate {{mvar|x}} and a prime {{mvar|p}} and is given by<ref>{{Apostol IANT}} pp.42–43</ref>

<math display="block">\operatorname{BG}_p(a_n;x) = \sum_{n=0}^\infty a_{p^n}x^n.</math>

===Dirichlet series generating functions (DGFs)===

[[Formal Dirichlet series]] are often classified as generating functions, although they are not strictly formal power series. The ''Dirichlet series generating function'' of a sequence {{math|''a''''n''}} is<ref name=W56>{{harvnb|Wilf|1994|p=56}}</ref>

<math display="block">\operatorname{DG}(a_n;s)=\sum _{n=1}^\infty \frac{a_n}{n^s}.</math>

The Dirichlet series generating function is especially useful when {{math|''a''''n''}} is a [[multiplicative function]], in which case it has an [[Euler product]] expression<ref name=W59>{{harvnb|Wilf|1994|p=59}}</ref> in terms of the function's Bell series

<math display="block">\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.</math>

If {{math|''a''''n''}} is a [[Dirichlet character]] then its Dirichlet series generating function is called a [[Dirichlet L-series|Dirichlet {{mvar|L}}-series]]. We also have a relation between the pair of coefficients in the [[Lambert series]] expansions above and their DGFs. Namely, we can prove that

<math display="block">[x^n] \operatorname{LG}(a_n; x) = b_n</math>

if and only if

<math display="block">\operatorname{DG}(a_n;s) \zeta(s) = \operatorname{DG}(b_n;s),</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]].<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |last3=Heath-Brown |first3=D.R |last4=Silverman |first4=J.H. |title=An Introduction to the Theory of Numbers|url=https://archive.org/details/introductiontoth00ghha_922|url-access=limited|publisher=Oxford University Press |page=[https://archive.org/details/introductiontoth00ghha_922/page/n357 339]|edition=6th |isbn=9780199219858 |year=2008}}</ref>

===Polynomial sequence generating functions===

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of [[binomial type]] are generated by

<math display="block">e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n</math>

where {{math|''p''''n''(''x'')}} is a sequence of polynomials and {{math|''f''(''t'')}} is a function of a certain form. [[Sheffer sequence]]s are generated in a similar way. See the main article [[generalized Appell polynomials]] for more information.

== Ordinary generating functions ==

=== Examples of generating functions for simple sequences ===

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the [[Poincaré polynomial]] and others.

A fundamental generating function is that of the constant sequence {{nowrap|1, 1, 1, 1, 1, 1, 1, 1, 1, ...}}, whose ordinary generating function is the [[Geometric_series#Closed-form_formula|geometric series]]

<math display="block">\sum_{n=0}^\infty x^n= \frac{1}{1-x}.</math>

The left-hand side is the [[Maclaurin series]] expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by {{math|1 − ''x''}}, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of {{math|''x''0}} are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the [[multiplicative inverse]] of {{math|1 − ''x''}} in the ring of power series.

Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution {{math|''x'' → ''ax''}} gives the generating function for the [[Geometric progression|geometric sequence]] {{math|1, ''a'', ''a''2, ''a''3, ...}} for any constant {{mvar|a}}:

<math display="block">\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.</math>

(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,

<math display="block">\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.</math>

One can also introduce regular gaps in the sequence by replacing {{mvar|x}} by some power of {{mvar|x}}, so for instance for the sequence {{nowrap|1, 0, 1, 0, 1, 0, 1, 0, ...}} (which skips over {{math|''x'', ''x''3, ''x''5, ...}}) one gets the generating function

<math display="block">\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.</math>

By squaring the initial generating function, or by finding the derivative of both sides with respect to {{mvar|x}} and making a change of running variable {{math|''n'' → ''n'' + 1}}, one sees that the coefficients form the sequence {{nowrap|1, 2, 3, 4, 5, ...}}, so one has

<math display="block">\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},</math>

and the third power has as coefficients the [[triangular number]]s {{nowrap|1, 3, 6, 10, 15, 21, ...}} whose term {{mvar|n}} is the [[binomial coefficient]] {{math|{{pars|s=150%|{{su|p=''n'' + 2|b=2|a=c}}}}}}, so that

<math display="block">\sum_{n=0}^\infty\binom{n+2}2 x^n= \frac{1}{(1-x)^3}.</math>

More generally, for any non-negative integer {{mvar|k}} and non-zero real value {{mvar|a}}, it is true that

<math display="block">\sum_{n=0}^\infty a^n\binom{n+k}k x^n= \frac{1}{(1-ax)^{k+1}}\,.</math>

Since

<math display="block">2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0 = 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2,</math>

one can find the ordinary generating function for the sequence {{nowrap|0, 1, 4, 9, 16, ...}} of [[square number]]s by linear combination of binomial-coefficient generating sequences:

<math display="block">G(n^2;x) = \sum_{n=0}^\infty n^2x^n = \frac{2}{(1-x)^3} - \frac{3}{(1-x)^2} + \frac{1}{1-x} = \frac{x(x+1)}{(1-x)^3}.</math>

We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the [[geometric series]] in the following form:

<math display="block">\begin{align}
G(n^2;x)
& = \sum_{n=0}^\infty n^2x^n \\[4px]
& = \sum_{n=0}^\infty n(n-1) x^n + \sum_{n=0}^\infty n x^n \\[4px]
& = x^2 D^2\left[\frac{1}{1-x}\right] + x D\left[\frac{1}{1-x}\right] \\[4px]
& = \frac{2 x^2}{(1-x)^3} + \frac{x}{(1-x)^2} =\frac{x(x+1)}{(1-x)^3}.
\end{align}</math>

By induction, we can similarly show for positive integers {{math|''m'' ≥ 1}} that<ref>{{cite journal|first1= Michael Z. | last1=Spivey | title=Combinatorial Sums and Finite Differences | year=2007 |journal = Discrete Math. |doi = 10.1016/j.disc.2007.03.052 | volume=307|number=24|pages=3130–3146|mr=2370116|doi-access=free }}</ref><ref>{{cite arXiv|first1=R. J. |last1=Mathar|year=2012|eprint=1207.5845|title=Yet another table of integrals|class=math.CA}} v4 eq. (0.4)</ref>

<math display="block">n^m = \sum_{j=0}^m \begin{Bmatrix} m \\ j \end{Bmatrix} \frac{n!}{(n-j)!}, </math>

where {{math|{{resize|150%|{}}{{su|p=''n''|b=''k''}}{{resize|150%|}<nowiki/>}}}} denote the [[Stirling numbers of the second kind]] and where the generating function

<math display="block">\sum_{n = 0}^\infty \frac{n!}{ (n-j)!} \, z^n = \frac{j! \cdot z^j}{(1-z)^{j+1}},</math>

so that we can form the analogous generating functions over the integral {{mvar|m}}th powers generalizing the result in the square case above. In particular, since we can write

<math display="block">\frac{z^k}{(1-z)^{k+1}} = \sum_{i=0}^k \binom{k}{i} \frac{(-1)^{k-i}}{(1-z)^{i+1}},</math>

we can apply a well-known finite sum identity involving the [[Stirling numbers]] to obtain that<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Table 265 in §6.1}} for finite sum identities involving the Stirling number triangles.</ref>

<math display="block">\sum_{n = 0}^\infty n^m z^n = \sum_{j=0}^m \begin{Bmatrix} m+1 \\ j+1 \end{Bmatrix} \frac{(-1)^{m-j} j!}{(1-z)^{j+1}}. </math>

=== Rational functions ===
{{Main|Linear recursive sequence}}
The ordinary generating function of a sequence can be expressed as a [[rational function]] (the ratio of two finite-degree polynomials) if and only if the sequence is a [[linear recursive sequence]] with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear [[finite difference equation]] with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive [[Binet's formula]] for the [[Fibonacci numbers]] via generating function techniques.

We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate ''quasi-polynomial'' sequences of the form <ref name="GFLECT">{{harvnb|Lando|2003|loc=§2.4}}</ref>

<math display="block">f_n = p_1(n) \rho_1^n + \cdots + p_\ell(n) \rho_\ell^n, </math>

where the reciprocal roots, <math>\rho_i \isin \mathbb{C}</math>, are fixed scalars and where {{math|''p''''i''(''n'')}} is a polynomial in {{mvar|n}} for all {{math|1 ≤ ''i'' ≤ ''ℓ''}}.

In general, [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard products]] of rational functions produce rational generating functions. Similarly, if

<math display="block">F(s, t) := \sum_{m,n \geq 0} f(m, n) w^m z^n</math>

is a bivariate rational generating function, then its corresponding ''diagonal generating function'',

<math display="block">\operatorname{diag}(F) := \sum_{n = 0}^\infty f(n, n) z^n,</math>

is ''algebraic''. For example, if we let<ref>Example from {{cite book |chapter=§6.3 |first1=Richard P. |last1=Stanley |first2=Sergey |last2=Fomin |title=Enumerative Combinatorics: Volume 2 |url=https://books.google.com/books?id=zg5wDqT6T-UC |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-78987-5 |series=Cambridge Studies in Advanced Mathematics |volume=62}}</ref>

<math display="block">F(s, t) := \sum_{i,j \geq 0} \binom{i+j}{i} s^i t^j = \frac{1}{1-s-t}, </math>

then this generating function's diagonal coefficient generating function is given by the well-known OGF formula

<math display="block">\operatorname{diag}(F) = \sum_{n = 0}^\infty \binom{2n}{n} z^n = \frac{1}{\sqrt{1-4z}}. </math>

This result is computed in many ways, including [[Cauchy's integral formula]] or [[contour integration]], taking complex [[residue (complex analysis)|residue]]s, or by direct manipulations of [[formal power series]] in two variables.

=== Operations on generating functions ===

==== Multiplication yields convolution ====
{{Main|Cauchy product}}
Multiplication of ordinary generating functions yields a discrete [[convolution]] (the [[Cauchy product]]) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general [[Euler–Maclaurin formula]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''an''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[Generating function#Convolution (Cauchy products)|section on convolutions]] in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.

==== Shifting sequence indices ====

For integers {{math|''m'' ≥ 1}}, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of {{math|⟨ ''g''''n'' − ''m'' ⟩}} and {{math|⟨ ''g''''n'' + ''m'' ⟩}}, respectively:

<math display="block">\begin{align}
& z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n \\[4px]
& \frac{G(z) - g_0 - g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m} = \sum_{n = 0}^\infty g_{n+m} z^n.
\end{align}</math>

==== Differentiation and integration of generating functions ====

We have the following respective power series expansions for the first derivative of a generating function and its integral:

<math display="block">\begin{align}
G'(z) & = \sum_{n = 0}^\infty (n+1) g_{n+1} z^n \\[4px]
z \cdot G'(z) & = \sum_{n = 0}^\infty n g_{n} z^n \\[4px]
\int_0^z G(t) \, dt & = \sum_{n = 1}^\infty \frac{g_{n-1}}{n} z^n.
\end{align}</math>

The differentiation–multiplication operation of the second identity can be repeated {{mvar|k}} times to multiply the sequence by {{math|''n''''k''}}, but that requires alternating between differentiation and multiplication. If instead doing {{mvar|k}} differentiations in sequence, the effect is to multiply by the {{mvar|k}}th [[falling factorial]]:

<math display="block"> z^k G^{(k)}(z) = \sum_{n = 0}^\infty n^\underline{k} g_n z^n = \sum_{n = 0}^\infty n (n-1) \dotsb (n-k+1) g_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

Using the [[Stirling numbers of the second kind]], that can be turned into another formula for multiplying by <math>n^k</math> as follows (see the main article on [[Generating function transformation#Derivative transformations|generating function transformations]]):

<math display="block"> \sum_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} z^j F^{(j)}(z) = \sum_{n = 0}^\infty n^k f_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the [[Generating function transformation#Derivative transformations|zeta series transformation]] and its generalizations defined as a derivative-based [[generating function transformation|transformation of generating functions]], or alternately termwise by and performing an [[Generating function transformation#Polylogarithm series transformations|integral transformation]] on the sequence generating function. Related operations of performing [[fractional calculus|fractional integration]] on a sequence generating function are discussed [[Generating function transformation#Fractional integrals and derivatives|here]].

==== Enumerating arithmetic progressions of sequences ====
In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}}, {{math|0 ≤ ''b'' < ''a''}}, and {{math|''a''}} and {{math|''b''}} are integers (see the [[generating function transformation|main article on transformations]]). For {{math|''a'' {{=}} 2}}, this is simply the familiar decomposition of a function into [[even and odd functions|even and odd parts]] (i.e., even and odd powers):

<math display="block">\begin{align}
\sum_{n = 0}^\infty f_{2n} z^{2n} &= \frac{F(z) + F(-z)}{2} \\[4px]
\sum_{n = 0}^\infty f_{2n+1} z^{2n+1} &= \frac{F(z) - F(-z)}{2}.
\end{align}</math>

More generally, suppose that {{math|''a'' ≥ 3}} and that {{math|''ωa'' {{=}} exp {{sfrac|2''πi''|''a''}}}} denotes the {{mvar|a}}th [[root of unity|primitive root of unity]]. Then, as an application of the [[discrete Fourier transform]], we have the formula<ref name="TAOCPV1">{{harvnb|Knuth|1997|loc=§1.2.9}}</ref>

<math display="block">\sum_{n = 0}^\infty f_{an+b} z^{an+b} = \frac{1}{a} \sum_{m=0}^{a-1} \omega_a^{-mb} F\left(\omega_a^m z\right).</math>

For integers {{math|''m'' ≥ 1}}, another useful formula providing somewhat ''reversed'' floored arithmetic progressions — effectively repeating each coefficient {{mvar|m}} times — are generated by the identity<ref>Solution to {{harvnb|Graham|Knuth|Patashnik|1994|p=569, exercise 7.36}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{\left\lfloor \frac{n}{m} \right\rfloor} z^n = \frac{1-z^m}{1-z} F(z^m) = \left(1 + z + \cdots + z^{m-2} + z^{m-1}\right) F(z^m).</math>

==={{math|''P''}}-recursive sequences and holonomic generating functions===

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>

<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''ci''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> spanned by the set of all of its derivatives is finite dimensional.

Since we can clear denominators if need be in the previous equation, we may assume that the functions, {{math|''ci''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form

<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''0}} and where the {{math|''ĉi''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.

====Examples====

The functions {{math|''e''''z''}}, {{math|log ''z''}}, {{math|cos ''z''}}, {{math|arcsin ''z''}}, <math>\sqrt{1 + z}</math>, the [[dilogarithm]] function {{math|Li2(''z'')}}, the [[generalized hypergeometric function]]s {{math|''pFq''(...; ...; ''z'')}} and the functions defined by the power series

<math display="block">\sum_{n = 0}^\infty \frac{z^n}{(n!)^2}</math>

and the non-convergent

<math display="block">\sum_{n = 0}^\infty n! \cdot z^n</math>

are all holonomic.

Examples of {{mvar|P}}-recursive sequences with holonomic generating functions include {{math|''f''''n'' ≔ {{sfrac|1|''n'' + 1}} {{pars|s=150%|{{su|p=2''n''|b=''n''|a=c}}}}}} and {{math|''f''''n'' ≔ {{sfrac|2''n''|''n''2 + 1}}}}, where sequences such as <math>\sqrt{n}</math> and {{math|log ''n''}} are ''not'' {{mvar|P}}-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as {{math|tan ''z''}}, {{math|sec ''z''}}, and [[Gamma function|{{math|Γ(''z'')}}]] are ''not'' holonomic functions.

====Software for working with ''{{mvar|P}}''-recursive sequences and holonomic generating functions====

Tools for processing and working with {{mvar|P}}-recursive sequences in ''[[Mathematica]]'' include the software packages provided for non-commercial use on the [https://www.risc.jku.at/research/combinat/software/ RISC Combinatorics Group algorithmic combinatorics software] site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>'''Guess'''</code> package for guessing ''{{mvar|P}}-recurrences'' for arbitrary input sequences (useful for [[experimental mathematics]] and exploration) and the <code>'''Sigma'''</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to {{mvar|P}}-recurrences involving generalized [[harmonic number]]s.<ref>{{cite journal|last1=Schneider|first1=C.|title=Symbolic Summation Assists Combinatorics|journal=Sem. Lothar. Combin.|date=2007|volume=56|pages=1–36 |url=http://www.emis.de/journals/SLC/wpapers/s56schneider.html}}</ref> Other packages listed on this particular RISC site are targeted at working with holonomic ''generating functions'' specifically.


=== Relation to discrete-time Fourier transform ===
{{Main|Discrete-time Fourier transform}}
When the series [[Absolute convergence|converges absolutely]],
<math display="block">G \left ( a_n; e^{-i \omega} \right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math>
is the discrete-time Fourier transform of the sequence {{math|''a''0, ''a''1, ...}}.

=== Asymptotic growth of a sequence ===
In calculus, often the growth rate of the coefficients of a power series can be used to deduce a [[radius of convergence]] for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the [[Asymptotic analysis|asymptotic growth]] of the underlying sequence.

For instance, if an ordinary generating function {{math|''G''(''a''''n''; ''x'')}} that has a finite radius of convergence of {{mvar|r}} can be written as

<math display="block">G(a_n; x) = \frac{A(x) + B(x) \left (1- \frac{x}{r} \right )^{-\beta}}{x^\alpha}</math>

where each of {{math|''A''(''x'')}} and {{math|''B''(''x'')}} is a function that is [[analytic function|analytic]] to a radius of convergence greater than {{mvar|r}} (or is [[Entire function|entire]]), and where {{math|''B''(''r'') ≠ 0}} then

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1}\left(\frac{1}{r}\right)^n \sim \frac{B(r)}{r^{\alpha}} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n = \frac{B(r)}{r^\alpha} \left(\!\!\binom{\beta}{n}\!\!\right)\left(\frac{1}{r}\right)^n\,,</math>
using the [[gamma function]], a [[binomial coefficient]], or a [[multiset coefficient]].

Often this approach can be iterated to generate several terms in an asymptotic series for {{math|''a''''n''}}. In particular,

<math display="block">G\left(a_n - \frac{B(r)}{r^\alpha} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n; x \right) = G(a_n; x) - \frac{B(r)}{r^\alpha} \left(1 - \frac{x}{r}\right)^{-\beta}\,.</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of {{mvar|A}}, {{mvar|B}}, {{mvar|α}}, {{mvar|β}}, and {{mvar|r}} to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is {{math|{{sfrac|''a''''n''|''n''!}}}} that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

==== Asymptotic growth of the sequence of squares ====
As derived above, the ordinary generating function for the sequence of squares is

<math display="block">G(n^2; x) = \frac{x(x+1)}{(1-x)^3}.</math>

With {{math|1=''r'' = 1}}, {{math|1=''α'' = −1}}, {{math|1=''β'' = 3}}, {{math|1=''A''(''x'') = 0}}, and {{math|1=''B''(''x'') = ''x'' + 1}}, we can verify that the squares grow as expected, like the squares:

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left (\frac{1}{r} \right)^n = \frac{1+1}{1^{-1}\,\Gamma(3)}\,n^{3-1} \left(\frac1 1\right)^n = n^2.</math>

==== Asymptotic growth of the Catalan numbers ====
{{Main|Catalan number}}

The ordinary generating function for the [[Catalan number]]s is

<math display="block">G(C_n; x) = \frac{1-\sqrt{1-4x}}{2x}.</math>

With {{math|1=''r'' = {{sfrac|1|4}}}}, {{math|1=''α'' = 1}}, {{math|1=''β'' = −{{sfrac|1|2}}}}, {{math|1=''A''(''x'') = {{sfrac|1|2}}}}, and {{math|1=''B''(''x'') = −{{sfrac|1|2}}}}, we can conclude that, for the Catalan numbers,

<math display="block">C_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left(\frac{1}{r} \right)^n = \frac{-\frac12}{\left(\frac14\right)^1 \Gamma\left(-\frac12\right)} \, n^{-\frac12-1} \left(\frac{1}{\,\frac14\,}\right)^n = \frac{4^n}{n^\frac32 \sqrt\pi}.</math>

=== Bivariate and multivariate generating functions ===
One can define generating functions in several variables for arrays with several indices. These are called '''multivariate generating functions''' or, sometimes, '''super generating functions'''. For two variables, these are often called '''bivariate generating functions'''.

For instance, since {{math|(1 + ''x'')''n''}} is the ordinary generating function for [[binomial coefficients]] for a fixed {{mvar|n}}, one may ask for a bivariate generating function that generates the binomial coefficients {{math|{{pars|s=150%|{{su|p=''n''|b=''k''|a=c}}}}}} for all {{mvar|k}} and {{mvar|n}}. To do this, consider {{math|(1 + ''x'')''n''}} itself as a sequence in {{mvar|n}}, and find the generating function in {{mvar|y}} that has these sequence values as coefficients. Since the generating function for {{math|''a''''n''}} is

<math display="block">\frac{1}{1-ay},</math>

the generating function for the binomial coefficients is:

<math display="block">\sum_{n,k} \binom{n}{k} x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}.</math>

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-order accurate]] power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the [[Jacobi-type continued fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>

<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
& = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''jn'' ≔ [''zn''] ''J''[∞](''z'')}}, in the previous equations correspond to matrix solutions of the equations

<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
\begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
\begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''0 ≡ ''k''0,0 {{=}} 1}}, {{math|''jn'' {{=}} ''k''0,''n''}} for {{math|''n'' ≥ 1}}, {{math|''k''''r'',''s'' {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by

<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''[∞](''z'')}}, expanded by

<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''Ph''(''z'')}} and {{math|''Qh''(''z'')}}, defined recursively by

<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv''h''(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''jn''}}, ''and'' for {{math|''Mh'' ≔ ab2 ⋯ ab''h'' + 1}} if {{math|''h'' ‖ ''M''''h''}} then we have the congruence

<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.

==== Examples ====

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''jn''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

:{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.

==Examples==

Generating functions for the sequence of [[square number]]s {{math|''a''''n'' {{=}} ''n''2}} are:

===Ordinary generating function===
<math display="block">G(n^2;x)=\sum_{n=0}^\infty n^2x^n = \frac{x(x+1)}{(1-x)^3}</math>

===Exponential generating function===
<math display="block">\operatorname{EG}(n^2;x)=\sum _{n=0}^\infty \frac{n^2x^n}{n!}=x(x+1)e^x</math>

===Lambert series===

As an example of a Lambert series identity not given in the [[Lambert series|main article]], we can show that for {{math|{{abs|''x''}}, {{abs|''xq''}} < 1}} we have that <ref>{{cite web|title=Lambert series identity|url=https://mathoverflow.net/q/140418 |website=Math Overflow|date=2017}}</ref>

<math display="block">\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n}, </math>

where we have the special case identity for the generating function of the [[divisor function]], {{math|''d''(''n'') ≡ ''σ''0(''n'')}}, given by

<math display="block">\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}. </math>

===Bell series===
<math display="block">\operatorname{BG}_p\left(n^2;x\right)=\sum_{n=0}^\infty \left(p^{n}\right)^2x^n=\frac{1}{1-p^2x}</math>

===Dirichlet series generating function===
<math display="block">\operatorname{DG}\left(n^2;s\right)=\sum_{n=1}^\infty \frac{n^2}{n^s}=\zeta(s-2),</math>

using the [[Riemann zeta function]].

The sequence {{mvar|ak}} generated by a [[Dirichlet series]] generating function (DGF) corresponding to:

<math display="block">\operatorname{DG}(a_k;s)=\zeta(s)^m</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]], has the ordinary generating function:

<math display="block">\sum_{k=1}^{k=n} a_k x^k = x + \binom{m}{1} \sum_{2 \leq a \leq n} x^{a} + \binom{m}{2}\underset{ab \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty} x^{ab} + \binom{m}{3}\underset{abc \leq n}{\sum_{a = 2}^\infty \sum_{c = 2}^\infty \sum_{b = 2}^\infty} x^{abc} + \binom{m}{4}\underset{abcd \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty \sum_{c = 2}^\infty \sum_{d = 2}^\infty} x^{abcd} + \cdots</math>

===Multivariate generating functions===
Multivariate generating functions arise in practice when calculating the number of [[contingency tables]] of non-negative integers with specified row and column totals. Suppose the table has {{mvar|r}} rows and {{mvar|c}} columns; the row sums are {{math|''t''1, ''t''2 ... ''tr''}} and the column sums are {{math|''s''1, ''s''2 ... ''sc''}}. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal| doi=10.1214/aos/1176343649| last=Good| first=I. J.| title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables| journal=[[Annals of Statistics]]| year=1986| volume=4| issue=6|pages=1159–1189| doi-access=free}}</ref> the number of such tables is the coefficient of

<math display="block">x_1^{t_1}\cdots x_r^{t_r}y_1^{s_1}\cdots y_c^{s_c}</math>

in

<math display="block">\prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.</math>

In the bivariate case, non-polynomial double sum examples of so-termed "''double''" or "''super''" generating functions of the form

<math display="block">G(w, z) := \sum_{m,n \geq 0} g_{m,n} w^m z^n</math>

include the following two-variable generating functions for the [[binomial coefficients]], the [[Stirling numbers]], and the [[Eulerian numbers]]:<ref>See the usage of these terms in {{harvnb|Graham|Knuth|Patashnik|1994|loc=§7.4}} on special sequence generating functions.</ref>

<math display="block">\begin{align}
e^{z+wz} & = \sum_{m,n \geq 0} \binom{n}{m} w^m \frac{z^n}{n!} \\[4px]
e^{w(e^z-1)} & = \sum_{m,n \geq 0} \begin{Bmatrix} n \\ m \end{Bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1}{(1-z)^w} & = \sum_{m,n \geq 0} \begin{bmatrix} n \\ m \end{bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1-w}{e^{(w-1)z}-w} & = \sum_{m,n \geq 0} \left\langle\begin{matrix} n \\ m \end{matrix} \right\rangle w^m \frac{z^n}{n!} \\[4px]
\frac{e^w-e^z}{w e^z-z e^w} &= \sum_{m,n \geq 0} \left\langle\begin{matrix} m+n+1 \\ m \end{matrix} \right\rangle \frac{w^m z^n}{(m+n+1)!}.
\end{align}</math>

==Applications==

===Various techniques: Evaluating sums and tackling other problems with generating functions===

====Example 1: A formula for sums of harmonic numbers====

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when {{math|''sn'' {{=}} Σ{{su|b=''k'' {{=}} 0|p=''n''}} ''ak''}}. We then know that {{math|''S''(''z'') {{=}} {{sfrac|''A''(''z'')|1 − ''z''}}}} for the corresponding ordinary generating functions.

For example, we can manipulate
<math display="block">s_n=\sum_{k=1}^{n} H_{k}\,,</math>
where {{math|''Hk'' {{=}} 1 + {{sfrac|1|2}} + ⋯ + {{sfrac|1|''k''}}}} are the [[harmonic number]]s. Let
<math display="block">H(z) = \sum_{n = 1}^\infty{H_n z^n}</math>
be the ordinary generating function of the harmonic numbers. Then
<math display="block">H(z) = \frac{1}{1-z}\sum_{n = 1}^\infty \frac{z^n}{n}\,,</math>
and thus
<math display="block">S(z) = \sum_{n = 1}^\infty{s_n z^n} = \frac{1}{(1-z)^2}\sum_{n = 1}^\infty \frac{z^n}{n}\,.</math>

Using
<math display="block">\frac{1}{(1-z)^2} = \sum_{n = 0}^\infty (n+1)z^n\,,</math>
[[Generating function#Convolution (Cauchy products)|convolution]] with the numerator yields
<math display="block">s_n = \sum_{k = 1}^{n} \frac{n+1-k}{k} = (n+1)H_n - n\,,</math>
which can also be written as
<math display="block">\sum_{k = 1}^{n}{H_k} = (n+1)(H_{n+1} - 1)\,.</math>

====Example 2: Modified binomial coefficient sums and the binomial transform====

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence {{math|⟨ ''fn'' ⟩}} we define the two sequences of sums
<math display="block">\begin{align}
s_n &:= \sum_{m=0}^n \binom{n}{m} f_m 3^{n-m} \\[4px]
\tilde{s}_n &:= \sum_{m=0}^n \binom{n}{m} (m+1)(m+2)(m+3) f_m 3^{n-m}\,,
\end{align}</math>
for all {{math|''n'' ≥ 0}}, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the [[binomial transform]] to write the generating function for the first sum as
<math display="block">S(z) = \frac{1}{1-3z} F\left(\frac{z}{1-3z}\right). </math>

Since the generating function for the sequence {{math|⟨ (''n'' + 1)(''n'' + 2)(''n'' + 3) ''fn'' ⟩}} is given by
<math display="block">6 F(z) + 18z F'(z) + 9z^2 F''(z) + z^3 F'''(z)</math>
we may write the generating function for the second sum defined above in the form
<math display="block">\tilde{S}(z) = \frac{6}{(1-3z)} F\left(\frac{z}{1-3z}\right)+\frac{18z}{(1-3z)^2} F'\left(\frac{z}{1-3z}\right)+\frac{9z^2}{(1-3z)^3} F''\left(\frac{z}{1-3z}\right)+\frac{z^3}{(1-3z)^4} F'''\left(\frac{z}{1-3z}\right). </math>

In particular, we may write this modified sum generating function in the form of
<math display="block">a(z) \cdot S(z) + b(z) \cdot z S'(z) + c(z) \cdot z^2 S''(z) + d(z) \cdot z^3 S'''(z), </math>
for {{math|''a''(''z'') {{=}} 6(1 − 3''z'')3}}, {{math|''b''(''z'') {{=}} 18(1 − 3''z'')3}}, {{math|''c''(''z'') {{=}} 9(1 − 3''z'')3}}, and {{math|''d''(''z'') {{=}} (1 − 3''z'')3}}, where {{math|(1 − 3''z'')3 {{=}} 1 − 9''z'' + 27''z''2 − 27''z''3}}.

Finally, it follows that we may express the second sums through the first sums in the following form:
<math display="block">\begin{align}
\tilde{s}_n & = [z^n]\left(6(1-3z)^3 \sum_{n = 0}^\infty s_n z^n + 18 (1-3z)^3 \sum_{n = 0}^\infty n s_n z^n + 9 (1-3z)^3 \sum_{n = 0}^\infty n(n-1) s_n z^n + (1-3z)^3 \sum_{n = 0}^\infty n(n-1)(n-2) s_n z^n\right) \\[4px]
& = (n+1)(n+2)(n+3) s_n - 9 n(n+1)(n+2) s_{n-1} + 27 (n-1)n(n+1) s_{n-2} - (n-2)(n-1)n s_{n-3}.
\end{align}</math>

====Example 3: Generating functions for mutually recursive sequences====

In this example, we reformulate a generating function example given in Section 7.3 of ''Concrete Mathematics'' (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted {{math|''Un''}}) to tile a 3-by-{{mvar|n}} rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, {{math|''Vn''}}, be defined as the number of ways to cover a 3-by-{{mvar|n}} rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a [[Closed-form expression|closed form]] formula for {{math|''Un''}} without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series

<math display="block">\begin{align}
U(z) = 1 + 3z^2 + 11 z^4 + 41 z^6 + \cdots, \\
V(z) = z + 4z^3 + 15 z^5 + 56 z^7 + \cdots.
\end{align}</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-{{mvar|n}} rectangle, we are able to express the following mutually dependent, or ''mutually recursive'', recurrence relations for our two sequences when {{math|''n'' ≥ 2}} defined as above where {{math|''U''0 {{=}} 1}}, {{math|''U''1 {{=}} 0}}, {{math|''V''0 {{=}} 0}}, and {{math|''V''1 {{=}} 1}}:
<math display="block">\begin{align}
U_n & = 2 V_{n-1} + U_{n-2} \\
V_n & = U_{n-1} + V_{n-2}.
\end{align}</math>

Since we have that for all integers {{math|''m'' ≥ 0}}, the index-shifted generating functions satisfy{{noteTag|Incidentally, we also have a corresponding formula when {{math|''m'' < 0}} given by
<math display="block">\sum_{n = 0}^\infty g_{n+m} z^n = \frac{G(z) - g_0 -g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m}\,.</math>}}
<math display="block">z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n\,,</math>
we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
<math display="block">\begin{align}
U(z) & = 2z V(z) + z^2 U(z) + 1 \\
V(z) & = z U(z) + z^2 V(z) = \frac{z}{1-z^2} U(z),
\end{align}</math>
which then implies by solving the system of equations (and this is the particular trick to our method here) that
<math display="block">U(z) = \frac{1-z^2}{1-4z^2+z^4} = \frac{1}{3-\sqrt{3}} \cdot \frac{1}{1-\left(2+\sqrt{3}\right) z^2} + \frac{1}{3 + \sqrt{3}} \cdot \frac{1}{1-\left(2-\sqrt{3}\right) z^2}. </math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that {{math|''U''2''n'' + 1 ≡ 0}} and that
<math display="block">U_{2n} = \left\lceil \frac{\left(2+\sqrt{3}\right)^n}{3-\sqrt{3}} \right\rceil\,, </math>
for all integers {{math|''n'' ≥ 0}}. We also note that the same shifted generating function technique applied to the second-order [[recurrence relation|recurrence]] for the [[Fibonacci numbers]] is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on [[rational functions]] given above.

===Convolution (Cauchy products)===

A discrete ''convolution'' of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see [[Cauchy product]]).

#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are ordinary generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow [z^n]C(z) = \sum_{k=0}^{n}{a_k b_{n-k}}</math>
#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are exponential generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow \left[\frac{z^n}{n!}\right]C(z) = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}</math>
#Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block">C(z) = F(z) G(z) H(z) \Leftrightarrow [z^n]C(z) = \sum_{j+k+ l=n} f_j g_k h_ l</math>
#Consider the {{mvar|m}}-fold convolution of a sequence with itself for some positive integer {{math|''m'' ≥ 1}} (see the example below for an application) <math display="block">C(z) = G(z)^m \Leftrightarrow [z^n]C(z) = \sum_{k_1+k_2+\cdots+k_m=n} g_{k_1} g_{k_2} \cdots g_{k_m}</math>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the [[probability generating function]], or ''pgf'', of a random variable {{mvar|Z}} is denoted by {{math|''GZ''(''z'')}}, then we can show that for any two random variables <ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=§8.3}}</ref>
<math display="block">G_{X+Y}(z) = G_X(z) G_Y(z)\,, </math>
if {{mvar|X}} and {{mvar|Y}} are independent. Similarly, the number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
<math display="block">C(z) = \frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}, </math>
and moreover, if we allow the {{mvar|n}} cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the [[partition function (mathematics)|partition function]] generating function expanded by the infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product of
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>

====Example: The generating function for the Catalan numbers====

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the [[Catalan numbers]], {{math|''Cn''}}. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product {{math|''x''0 · ''x''1 ·⋯· ''xn''}} so that the order of multiplication is completely specified. For example, {{math|''C''2 {{=}} 2}} which corresponds to the two expressions {{math|''x''0 · (''x''1 · ''x''2)}} and {{math|(''x''0 · ''x''1) · ''x''2}}. It follows that the sequence satisfies a recurrence relation given by
<math display="block">C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k} + \delta_{n,0} = C_0 C_{n-1} + C_1 C_{n-2} + \cdots + C_{n-1} C_0 + \delta_{n,0}\,,\quad n \geq 0\,, </math>
and so has a corresponding convolved generating function, {{math|''C''(''z'')}}, satisfying
<math display="block">C(z) = z \cdot C(z)^2 + 1\,.</math>

Since {{math|''C''(0) {{=}} 1 ≠ ∞}}, we then arrive at a formula for this generating function given by
<math display="block">C(z) = \frac{1-\sqrt{1-4z}}{2z} = \sum_{n = 0}^\infty \frac{1}{n+1}\binom{2n}{n} z^n\,.</math>

Note that the first equation implicitly defining {{math|''C''(''z'')}} above implies that
<math display="block">C(z) = \frac{1}{1-z \cdot C(z)} \,, </math>
which then leads to another "simple" (of form) continued fraction expansion of this generating function.

====Example: Spanning trees of fans and convolutions of convolutions====

A ''fan of order {{mvar|n}}'' is defined to be a graph on the vertices {{math|{0, 1, ..., ''n''}<nowiki/>}} with {{math|2''n'' − 1}} edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other {{mvar|n}} vertices, and vertex <math>k</math> is connected by a single edge to the next vertex {{math|''k'' + 1}} for all {{math|1 ≤ ''k'' < ''n''}}.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Example 6 in §7.3}} for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.</ref> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A [[spanning tree]] is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees {{math|''fn''}} of a fan of order {{mvar|n}} are possible for each {{math|''n'' ≥ 1}}.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when {{math|''n'' {{=}} 4}}, we have that {{math|''f''4 {{=}} 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 {{=}} 21}}, which is a sum over the {{mvar|m}}-fold convolutions of the sequence {{math|''gn'' {{=}} ''n'' {{=}} [''zn''] {{sfrac|''z''|(1 − ''z'')2}}}} for {{math|''m'' ≔ 1, 2, 3, 4}}. More generally, we may write a formula for this sequence as
<math display="block">f_n = \sum_{m > 0} \sum_{\scriptstyle k_1+k_2+\cdots+k_m=n\atop\scriptstyle k_1, k_2, \ldots,k_m > 0} g_{k_1} g_{k_2} \cdots g_{k_m}\,, </math>
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
<math display="block">F(z) = G(z) + G(z)^2 + G(z)^3 + \cdots = \frac{G(z)}{1-G(z)} = \frac{z}{(1-z)^2-z} = \frac{z}{1-3z+z^2}\,,</math>
from which we are able to extract an exact formula for the sequence by taking the [[partial fraction expansion]] of the last generating function.

===Implicit generating functions and the Lagrange inversion formula===
{{expand section|This section needs to be added to the list of techniques with generating functions|date=April 2017}}

One often encounters generating functions specified by a functional equation, instead of an explicit specification. For example, the generating function {{math|''T(z)''}} for the number of binary trees on {{math|''n''}} nodes (leaves included) satisfies

<math display = "block">T(z) = z\left(1+T(z)^2\right)</math>

The Lagrange Inversion Theorem is a tool used to explicitly evaluate solutions to such equations.

{{math theorem|Lagrange Inversion Formula| Let <math display = "inline"> \phi(z) \in C[[z]]</math> be a formal power series with a non-zero constant term. Then the functional equation
<math display = "block">T(z) = z \phi(T(z))</math>
admits a unique solution in <math display = "inline">T(z) \in C[[z]]</math>, which satisfies

<math mode = "block"> [z^n] T(z) = [z^{n-1}] \frac{1}{n} (\phi(z))^n </math>

where the notation <math mode = "inline">[z^n] F(z)</math> returns the coefficient of <math mode = "inline">z^n</math> in <math mode = "\inline">F(z)</math>.
}}

Applying the above theorem to our functional equation yields (with <math display = "inline">\phi(z) = 1+z^2</math>):

<math display = "block"> [z^n]T(z) = [z^{n-1}] \frac{1}{n} (1+z^2)^n </math>

Via the binomial theorem expansion, for even <math mode = "inline">n</math>, the formula returns <math mode = "inline">0</math>. This is expected as one can prove that the number of leaves of a binary tree are one more than the number of its internal nodes, so the total sum should always be an odd number. For odd <math mode = "inline">n</math>, however, we get

<math mode = "block">[z^{n-1}] \frac{1}{n} (1+z^2)^n = \frac{1}{n} \dbinom{n}{\frac{n+1}{2}} </math>

The expression becomes much neater if we let <math mode = "inline">n</math> be the number of internal nodes: Now the expression just becomes the <math mode = "inline">n</math>th Catalan number.

===Introducing a free parameter (snake oil method)===
Sometimes the sum {{math|''sn''}} is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have {{mvar|n}} as limit in the summation. When n does not appear explicitly in the summation, we may consider {{mvar|n}} as a "free" parameter and treat {{math|''sn''}} as a coefficient of {{math|''F''(''z'') {{=}} Σ ''sn'' ''zn''}}, change the order of the summations on {{mvar|n}} and {{mvar|k}}, and try to compute the inner sum.

For example, if we want to compute
<math display="block">s_n = \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\,, \quad m,n \in \mathbb{N}_0\,,</math>
we can treat {{mvar|n}} as a "free" parameter, and set
<math display="block">F(z) = \sum_{n = 0}^\infty{\left( \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\right) }z^n\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">F(z) = \sum_{k = 0}^\infty{\binom{2k}{k}\frac{(-1)^k}{k+1} z^{-k}}\sum_{n = 0}^\infty{\binom{n+k}{m+2k} z^{n+k}}\,.</math>

Now the inner sum is {{math|{{sfrac|''z''''m'' + 2''k''|(1 − ''z'')''m'' + 2''k'' + 1}}}}. Thus
<math display="block">\begin{align} F(z)
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{\frac{1}{k+1}\binom{2k}{k}\left(\frac{-z}{(1-z)^2}\right)^k} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{C_k\left(\frac{-z}{(1-z)^2}\right)^k} &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^2}}}{\frac{-2z}{(1-z)^2}} \\[4px]
&= \frac{-z^{m-1}}{2(1-z)^{m-1}}\left(1-\frac{1+z}{1-z}\right) \\[4px]
&= \frac{z^m}{(1-z)^m} = z\frac{z^{m-1}}{(1-z)^m}\,.
\end{align}</math>

Then we obtain
<math display="block">s_n = \begin{cases}
\displaystyle\binom{n-1}{m-1} & \text{for } m \geq 1 \,, \\ {}
[n = 0] & \text{for } m = 0\,.
\end{cases}</math>

It is instructive to use the same method again for the sum, but this time take {{mvar|m}} as the free parameter instead of {{mvar|n}}. We thus set
<math display="block">G(z) = \sum_{m = 0}^\infty\left( \sum_{k = 0}^\infty \binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1} \right) z^m\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">G(z) = \sum_{k = 0}^\infty \binom{2k}{k}\frac{(-1)^k}{k+1} z^{-2k} \sum_{m = 0}^\infty \binom{n+k}{m+2k} z^{m+2k}\,.</math>

Now the inner sum is {{math|(1 + ''z'')''n'' + ''k''}}. Thus
<math display="block">\begin{align} G(z)
&= (1+z)^n \sum_{k = 0}^\infty \frac{1}{k+1}\binom{2k}{k}\left(\frac{-(1+z)}{z^2}\right)^k \\[4px]
&= (1+z)^n \sum_{k = 0}^\infty C_k \,\left(\frac{-(1+z)}{z^2}\right)^k &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= (1+z)^n \,\frac{1-\sqrt{1+\frac{4(1+z)}{z^2}}}{\frac{-2(1+z)}{z^2}} \\[4px]
&= (1+z)^n \,\frac{z^2-z\sqrt{z^2+4+4z}}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{z^2-z(z+2)}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{-2z}{-2(1+z)} = z(1+z)^{n-1}\,.
\end{align}</math>

Thus we obtain
<math display="block">s_n = \left[z^m\right] z(1+z)^{n-1} = \left[z^{m-1}\right] (1+z)^{n-1} = \binom{n-1}{m-1}\,,</math>
for {{math|''m'' ≥ 1}} as before.

===Generating functions prove congruences===
We say that two generating functions (power series) are congruent modulo {{mvar|m}}, written {{math|''A''(''z'') ≡ ''B''(''z'') (mod ''m'')}} if their coefficients are congruent modulo {{mvar|m}} for all {{math|''n'' ≥ 0}}, i.e., {{math|''an'' ≡ ''bn'' (mod ''m'')}} for all relevant cases of the integers {{mvar|n}} (note that we need not assume that {{mvar|m}} is an integer here—it may very well be polynomial-valued in some indeterminate {{mvar|x}}, for example). If the "simpler" right-hand-side generating function, {{math|''B''(''z'')}}, is a rational function of {{mvar|z}}, then the form of this sequence suggests that the sequence is [[periodic function|eventually periodic]] modulo fixed particular cases of integer-valued {{math|''m'' ≥ 2}}. For example, we can prove that the [[Euler numbers]],
<math display="block">\langle E_n \rangle = \langle 1, 1, 5, 61, 1385, \ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle \pmod{3}\,,</math>
satisfy the following congruence modulo 3:<ref>{{harvnb|Lando|2003|loc=§5}}</ref>
<math display="block">\sum_{n = 0}^\infty E_n z^n = \frac{1-z^2}{1+z^2} \pmod{3}\,. </math>

One useful method of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers {{math|''pk''}}) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by {{mvar|J}}-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's ''Lectures on Generating Functions'' as follows:
{{math theorem | name = Theorem: congruences for series generated by expansions of continued fractions
| math_statement = Suppose that the generating function {{math|''A''(''z'')}} is represented by an infinite [[continued fraction]] of the form
<math display="block">A(z) = \cfrac{1}{1-c_1z - \cfrac{p_1z^2}{1-c_2z - \cfrac{p_2 z^2}{1-c_3z - {\ddots}}}}</math>
and that {{math|''Ap''(''z'')}} denotes the {{mvar|p}}th convergent to this continued fraction expansion defined such that {{math|''an'' {{=}} [''zn''] ''Ap''(''z'')}} for all {{math|0 ≤ ''n'' < 2''p''}}. Then:

# the function {{math|''Ap''(''z'')}} is rational for all {{math|''p'' ≥ 2}} where we assume that one of divisibility criteria of {{math|''p'' {{!}} ''p''1, ''p''1''p''2, ''p''1''p''2''p''3}} is met, that is, {{math|''p'' {{!}} ''p''1''p''2⋯''p''''k''}} for some {{math|''k'' ≥ 1}}; and
# if the integer {{mvar|p}} divides the product {{math|''p''1''p''2⋯''p''''k''}}, then we have {{math|''A''(''z'') ≡ ''Ak''(''z'') (mod ''p'')}}.}}

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the [[Stirling numbers of the first kind]] and for the [[partition function (mathematics)|partition function {{math|''p''(''n'')}}]] which show the versatility of generating functions in tackling problems involving [[integer sequences]].

====The Stirling numbers modulo small integers====

The [[Stirling numbers of the first kind#Congruences|main article]] on the Stirling numbers generated by the finite products
<math display="block">S_n(x) := \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k = x(x+1)(x+2) \cdots (x+n-1)\,,\quad n \geq 1\,, </math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference ''Generatingfunctionology''.
We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<math display="block">S_n(x) = [x(x+1)] \cdot [x(x+1)] \cdots = x^{\left\lceil \frac{n}{2} \right\rceil} (x+1)^{\left\lfloor \frac{n}{2} \right\rfloor}\,, </math>

which implies that the parity of these [[Stirling numbers]] matches that of the binomial coefficient

<math display="block">\begin{bmatrix} n \\ k \end{bmatrix} \equiv \binom{\left\lfloor \frac{n}{2} \right\rfloor}{k - \left\lceil \frac{n}{2} \right\rceil} \pmod{2}\,, </math>

and consequently shows that {{math|{{resize|150%|[}}{{su|p=''n''|b=''k''|a=c}}{{resize|150%|]}}}} is even whenever {{math|''k'' < ⌊ {{sfrac|''n''|2}} ⌋}}.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that
<math display="block">\begin{align}
\begin{bmatrix} n \\ m \end{bmatrix} & \equiv
[x^m] \left(
x^{\left\lceil \frac{n}{3} \right\rceil} (x+1)^{\left\lceil \frac{n-1}{3} \right\rceil}
(x+2)^{\left\lfloor \frac{n}{3} \right\rfloor}
\right) && \pmod{3} \\
& \equiv
\sum_{k=0}^{m} \begin{pmatrix} \left\lceil \frac{n-1}{3} \right\rceil \\ k \end{pmatrix}
\begin{pmatrix} \left\lfloor \frac{n}{3} \right\rfloor \\ m-k - \left\lceil \frac{n}{3} \right\rceil \end{pmatrix} \times
2^{\left\lceil \frac{n}{3} \right\rceil + \left\lfloor \frac{n}{3} \right\rfloor -(m-k)} && \pmod{3}\,.
\end{align}</math>

====Congruences for the partition function====

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that ''the'' [[partition function (number theory)|partition function]] {{math|''p''(''n'')}} is generated by the reciprocal infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product (or {{mvar|z}}-Pochhammer product as the case may be) given by
<math display="block">\begin{align}
\sum_{n = 0}^\infty p(n) z^n & = \frac{1}{\left(1-z\right)\left(1-z^2\right)\left(1-z^3\right) \cdots} \\[4pt]
& = 1 + z + 2z^2 + 3 z^3 + 5z^4 + 7z^5 + 11z^6 + \cdots.
\end{align}</math>

This partition function satisfies many known [[Ramanujan's congruences|congruence properties]], which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<ref>{{harvnb|Hardy|Wright|Heath-Brown|Silverman|2008|loc=§19.12}}</ref>
<math display="block">\begin{align}
p(5m+4) & \equiv 0 \pmod{5} \\
p(7m+5) & \equiv 0 \pmod{7} \\
p(11m+6) & \equiv 0 \pmod{11} \\
p(25m+24) & \equiv 0 \pmod{5^2}\,.
\end{align}</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function
<math display=block>\frac{1}{(1-z)^5} = \sum_{i=0}^\infty \binom{4+i}{4}z^i\,,</math>
all of the coefficients are divisible by 5 except for those which correspond to the powers {{math|1, ''z''5, ''z''10, ...}} and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
<math display="block">\frac{1}{(1-z)^5} \equiv \frac{1}{1-z^5} \pmod{5}\,,</math>
or equivalently
<math display="block"> \frac{1-z^5}{(1-z)^5} \equiv 1 \pmod{5}\,.</math>
It follows that
<math display="block">\frac{\left(1-z^5\right)\left(1-z^{10}\right)\left(1-z^{15}\right) \cdots }{\left((1-z)\left(1-z^2\right)\left(1-z^3\right) \cdots \right)^5} \equiv 1 \pmod{5}\,. </math>

Using the infinite product expansions of
<math display="block">z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(1-z\right)\left(1-z^2\right) \cdots } =
z \cdot \left((1-z)\left(1-z^2\right) \cdots \right)^4 \times \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(\left(1-z\right)\left(1-z^2\right) \cdots \right)^5}\,,</math>
it can be shown that the coefficient of {{math|''z''5''m'' + 5}} in {{math|''z'' · ((1 − ''z'')(1 − ''z''2)⋯)4}} is divisible by 5 for all {{mvar|m}}.<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M.|title=An Introduction to the Theory of Numbers}} p.288, Th.361</ref> Finally, since
<math display="block">\begin{align}
\sum_{n = 1}^\infty p(n-1) z^n & = \frac{z}{(1-z)\left(1-z^2\right) \cdots} \\[6px]
& = z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{(1-z)\left(1-z^2\right) \cdots } \times \left(1+z^5+z^{10}+\cdots\right)\left(1+z^{10}+z^{20}+\cdots\right) \cdots
\end{align}</math>
we may equate the coefficients of {{math|''z''5''m'' + 5}} in the previous equations to prove our desired congruence result, namely that {{math|''p''(5''m'' + 4) ≡ 0 (mod 5)}} for all {{math|''m'' ≥ 0}}.

===Transformations of generating functions===
There are a number of transformations of generating functions that provide other applications (see the [[generating function transformation|main article]]). A transformation of a sequence's ''ordinary generating function'' (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see [[Generating function transformation#Integral Transformations|integral transformations]]) or weighted sums over the higher-order derivatives of these functions (see [[Generating function transformation#Derivative Transformations|derivative transformations]]).

Generating function transformations can come into play when we seek to express a generating function for the sums

<math display="block">s_n := \sum_{m=0}^n \binom{n}{m} C_{n,m} a_m, </math>

in the form of {{math|''S''(''z'') {{=}} ''g''(''z'') ''A''(''f''(''z''))}} involving the original sequence generating function. For example, if the sums are
<math display="block">s_n := \sum_{k = 0}^\infty \binom{n+k}{m+2k} a_k \,</math>
then the generating function for the modified sum expressions is given by<ref>{{harvnb|Graham|Knuth|Patashnik|1994|p=535, exercise 5.71}}</ref>
<math display="block">S(z) = \frac{z^m}{(1-z)^{m+1}} A\left(\frac{z}{(1-z)^2}\right)</math>
(see also the [[binomial transform]] and the [[Stirling transform]]).

There are also integral formulas for converting between a sequence's OGF, {{math|''F''(''z'')}}, and its exponential generating function, or EGF, {{math|''F̂''(''z'')}}, and vice versa given by

<math display="block">\begin{align}
F(z) &= \int_0^\infty \hat{F}(tz) e^{-t} \, dt \,, \\[4px]
\hat{F}(z) &= \frac{1}{2\pi} \int_{-\pi}^\pi F\left(z e^{-i\vartheta}\right) e^{e^{i\vartheta}} \, d\vartheta \,,
\end{align}</math>

provided that these integrals converge for appropriate values of {{mvar|z}}.

===Other applications===
Generating functions are used to:

* Find a [[closed formula]] for a sequence given in a recurrence relation. For example, consider [[Fibonacci number#Generating function|Fibonacci numbers]].
* Find [[recurrence relation]]s for sequences—the form of a generating function may suggest a recurrence formula.
* Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.
* Explore the asymptotic behaviour of sequences.
* Prove identities involving sequences.
* Solve [[enumeration]] problems in [[combinatorics]] and encoding their solutions. [[Rook polynomial]]s are an example of an application in combinatorics.
* Evaluate infinite sums.

==Other generating functions==

===Examples===

Examples of [[polynomial sequence]]s generated by more complex generating functions include:

* [[Appell polynomials]]
* [[Chebyshev polynomials]]
* [[Difference polynomials]]
* [[Generalized Appell polynomials]]
* [[Q-difference polynomial|{{mvar|q}}-difference polynomials]]

Other sequences generated by more complex generating functions:

* Double exponential generating functions. For example: [https://oeis.org/search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search Aitken's Array: Triangle of Numbers]
* Hadamard products of generating functions and diagonal generating functions, and their corresponding [[generating function transformation#Hadamard products and diagonal generating functions|integral transformations]]

===Convolution polynomials===

Knuth's article titled "''Convolution Polynomials''"<ref>{{cite journal|last1=Knuth|first1=D. E.|title=Convolution Polynomials|journal=Mathematica J.|date=1992|volume=2|pages=67–78|arxiv=math/9207221|bibcode=1992math......7221K}}</ref> defines a generalized class of ''convolution polynomial'' sequences by their special generating functions of the form
<math display="block">F(z)^x = \exp\bigl(x \log F(z)\bigr) = \sum_{n = 0}^\infty f_n(x) z^n,</math>
for some analytic function {{mvar|F}} with a power series expansion such that {{math|''F''(0) {{=}} 1}}.

We say that a family of polynomials, {{math|''f''0, ''f''1, ''f''2, ...}}, forms a ''convolution family'' if {{math|[[Degree of a polynomial|deg]] ''fn'' ≤ ''n''}} and if the following convolution condition holds for all {{mvar|x}}, {{mvar|y}} and for all {{math|''n'' ≥ 0}}:
<math display="block">f_n(x+y) = f_n(x) f_0(y) + f_{n-1}(x) f_1(y) + \cdots + f_1(x) f_{n-1}(y) + f_0(x) f_n(y). </math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

* The sequence {{math|''n''! · ''fn''(''x'')}} is of [[binomial type]]
* Special values of the sequence include {{math|''fn''(1) {{=}} [''zn''] ''F''(''z'')}} and {{math|''fn''(0) {{=}} ''δ''''n'',0}}, and
* For arbitrary (fixed) <math>x, y, t \isin \mathbb{C}</math>, these polynomials satisfy convolution formulas of the form
<math display="block">\begin{align}
f_n(x+y) & = \sum_{k=0}^n f_k(x) f_{n-k}(y) \\
f_n(2x) & = \sum_{k=0}^n f_k(x) f_{n-k}(x) \\
xn f_n(x+y) & = (x+y) \sum_{k=0}^n k f_k(x) f_{n-k}(y) \\
\frac{(x+y) f_n(x+y+tn)}{x+y+tn} & = \sum_{k=0}^n \frac{x f_k(x+tk)}{x+tk} \frac{y f_{n-k}(y+t(n-k))}{y+t(n-k)}.
\end{align}</math>

For a fixed non-zero parameter <math>t \isin \mathbb{C}</math>, we have modified generating functions for these convolution polynomial sequences given by
<math display="block">\frac{z F_n(x+tn)}{(x+tn)} = \left[z^n\right] \mathcal{F}_t(z)^x, </math>
where {{math|𝓕''t''(''z'')}} is implicitly defined by a [[functional equation]] of the form {{math|𝓕''t''(''z'') {{=}} ''F''(''x''𝓕''t''(''z'')''t'')}}. Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, {{math|⟨ ''fn''(''x'') ⟩}} and {{math|⟨ ''gn''(''x'') ⟩}}, with respective corresponding generating functions, {{math|''F''(''z'')''x''}} and {{math|''G''(''z'')''x''}}, then for arbitrary {{mvar|t}} we have the identity
<math display="block">\left[z^n\right] \left(G(z) F\left(z G(z)^t\right)\right)^x = \sum_{k=0}^n F_k(x) G_{n-k}(x+tk). </math>

Examples of convolution polynomial sequences include the ''binomial power series'', {{math|𝓑''t''(''z'') {{=}} 1 + ''z''𝓑''t''(''z'')''t''}}, so-termed ''tree polynomials'', the [[Bell numbers]], {{math|''B''(''n'')}}, the [[Laguerre polynomials]], and the [[Stirling polynomial|Stirling convolution polynomials]].

===Tables of special generating functions===

An initial listing of special mathematical series is found [[List of mathematical series|here]]. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of ''Concrete Mathematics'' and in Section 2.5 of Wilf's ''Generatingfunctionology''. Other special generating functions of note include the entries in the next table, which is by no means complete.<ref>See also the ''1031 Generating Functions'' found in {{cite thesis |first=Simon |last=Plouffe |title=Approximations de séries génératrices et quelques conjectures |trans-title=Approximations of generating functions and a few conjectures |year=1992 |type=Masters |publisher=Université du Québec à Montréal |language=fr |arxiv=0911.4975}}</ref>

{{expand section|Lists of special and special sequence generating functions. The next table is a start|date=April 2017}}

:{| class="wikitable"
|-
! Formal power series !! Generating-function formula !! Notes
|-
| <math>\sum_{n = 0}^\infty \binom{m+n}{n} \left(H_{n+m}-H_m\right) z^n</math> || <math>\frac{1}{(1-z)^{m+1}} \ln \frac{1}{1-z}</math> || <math>H_n</math> is a first-order [[harmonic number]]
|-
| <math>\sum_{n = 0}^\infty B_n \frac{z^n}{n!}</math> || <math>\frac{z}{e^z-1}</math> || <math>B_n</math> is a [[Bernoulli number]]
|-
| <math>\sum_{n = 0}^\infty F_{mn} z^n</math> || <math>\frac{F_m z}{1-(F_{m-1}+F_{m+1})z+(-1)^m z^2}</math> || <math>F_n</math> is a [[Fibonacci number]] and <math>m \in \mathbb{Z}^{+}</math>
|-
| <math>\sum_{n = 0}^\infty \left\{\begin{matrix} n \\ m \end{matrix} \right\} z^n</math> || <math>(z^{-1})^{\overline{-m}} = \frac{z^m}{(1-z)(1-2z)\cdots(1-mz)}</math> || <math>x^{\overline{n}}</math> denotes the [[rising factorial]], or [[Pochhammer symbol]] and some integer <math>m \geq 0</math>
|-
| <math>\sum_{n = 0}^\infty \left[\begin{matrix} n \\ m \end{matrix} \right] z^n</math> || <math>z^{\overline{m}} = z(z+1) \cdots (z+m-1)</math>
|-
| <math>\sum_{n = 1}^\infty \frac{(-1)^{n-1}4^n (4^n-2) B_{2n} z^{2n}}{(2n) \cdot (2n)!}</math> || <math>\ln \frac{\tan(z)}{z}</math>
|-
| <math>\sum_{n = 0}^\infty \frac{(1/2)^{\overline{n}} z^{2n}}{(2n+1) \cdot n!}</math> || <math>z^{-1} \arcsin(z)</math>
|-
| <math>\sum_{n = 0}^\infty H_n^{(s)} z^n</math> || <math>\frac{\operatorname{Li}_s(z)}{1-z}</math> || <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]] function and <math>H_n^{(s)}</math> is a generalized [[harmonic number]] for <math>\Re(s) > 1</math>
|-
| <math>\sum_{n = 0}^\infty n^m z^n</math> || <math>\sum_{0 \leq j \leq m} \left\{\begin{matrix} m \\ j \end{matrix} \right\} \frac{j! \cdot z^j}{(1-z)^{j+1}}</math> || <math>\left\{\begin{matrix} n \\ m \end{matrix} \right\}</math> is a [[Stirling number of the second kind]] and where the individual terms in the expansion satisfy <math>\frac{z^i}{(1-z)^{i+1}} = \sum_{k=0}^{i} \binom{i}{k} \frac{(-1)^{k-i}}{(1-z)^{k+1}}</math>
|-
| <math>\sum_{k < n} \binom{n-k}{k} \frac{n}{n-k} z^k</math> || <math>\left(\frac{1+\sqrt{1+4z}}{2}\right)^n + \left(\frac{1-\sqrt{1+4z}}{2}\right)^n</math> ||
|-
| <math>\sum_{n_1, \ldots, n_m \geq 0} \min(n_1, \ldots, n_m) z_1^{n_1} \cdots z_m^{n_m}</math> || <math>\frac{z_1 \cdots z_m}{(1-z_1) \cdots (1-z_m) (1-z_1 \cdots z_m)}</math> || The two-variable case is given by <math>M(w, z) := \sum_{m,n \geq 0} \min(m, n) w^m z^n = \frac{wz}{(1-w)(1-z)(1-wz)}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{s}{n} z^n</math> || <math>(1+z)^s</math> || <math>s \in \mathbb{C}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{n}{k} z^n</math> || <math>\frac{z^k}{(1-z)^{k+1}}</math> || <math>k \in \mathbb{N}</math>
|-
|<math>\sum_{n = 1}^\infty \log{(n)} z^n</math>||<math>\left.-\frac{\partial}{\partial s}\operatorname{{Li}_s(z)}\right|_{s=0}</math>||
|}

== History ==
[[George Pólya]] writes in ''[[Mathematics and plausible reasoning]]'':
<blockquote>''The name "generating function" is due to [[Laplace]]. Yet, without giving it a name, [[Euler]] used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the [[Number theory|Theory of Numbers]].''</blockquote>

==See also==
* [[Moment-generating function]]
* [[Probability-generating function]]
* [[Generating function transformation]]
* [[Stanley's reciprocity theorem]]
* Applications to [[Partition (number theory)]]
* [[Combinatorial principles]]
* [[Cyclic sieving]]
* [[Z-transform]]
* [[Umbral calculus]]

==Notes==
{{noteFoot}}

==References==
{{reflist}}

===Citations===
*{{cite book |first=Martin |last=Aigner |title=A Course in Enumeration |url=https://books.google.com/books?id=pPEJcu93dzAC |date=2007 |publisher=Springer |isbn=978-3-540-39035-0 |series=Graduate Texts in Mathematics |volume=238 }}
* {{cite journal |title=On the foundations of combinatorial theory. VI. The idea of generating function |last1=Doubilet |first1=Peter |last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | last3=Stanley | first3=Richard | author3-link=Richard P. Stanley | journal=Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability |volume=2 |pages=267–318 |year=1972 | zbl=0267.05002 | url=http://projecteuclid.org/euclid.bsmsp/1200514223 }} Reprinted in {{cite book | last=Rota | first=Gian-Carlo | author-link=Gian-Carlo Rota | others=With the collaboration of P. Doubilet, C. Greene, D. Kahaner, [[Andrew Odlyzko|A. Odlyzko]] and [[Richard P. Stanley|R. Stanley]] | title=Finite Operator Calculus | chapter=3. The idea of generating function | pages=83–134 | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 | zbl=0328.05007 }}
* {{cite book | last1 = Flajolet | first1 = Philippe | author-link1 = Philippe Flajolet | last2 = Sedgewick | first2 = Robert | author-link2 = Robert Sedgewick (computer scientist) | title = Analytic Combinatorics | title-link= Analytic Combinatorics | year = 2009 | publisher = Cambridge University Press | isbn = 978-0-521-89806-5 | zbl=1165.05001 }}
* {{cite book | last1 = Goulden | first1 = Ian P. | last2 = Jackson | first2 = David M. | author-link2 = David M. Jackson | title = Combinatorial Enumeration | year = 2004 | publisher = [[Dover Publications]] | isbn = 978-0486435978 }}
* {{cite book |title=[[Concrete Mathematics|Concrete Mathematics. A foundation for computer science]] |edition=2nd |year=1994 |publisher=Addison-Wesley |isbn=0-201-55802-5 |chapter=Chapter 7: Generating Functions |pages=320–380| zbl=0836.00001 |first1 = Ronald L. |last1=Graham |first2 = Donald E. |last2=Knuth |first3=Oren |last3=Patashnik |author-link1=Ronald Graham |author-link2=Donald Knuth |author-link3=Oren Patashnik }}
*{{cite book |first=Sergei K. |last=Lando |title=Lectures on Generating Functions |url=https://books.google.com/books?id=A6_4AwAAQBAJ |date=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3481-7 }}
* {{cite book | last=Wilf | first=Herbert S. | author-link=Herbert Wilf | title=Generatingfunctionology | edition=2nd | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 | url=http://www.math.upenn.edu/%7Ewilf/DownldGF.html }}

==External links==
* [http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf "Introduction To Ordinary Generating Functions"] by Mike Zabrocki, York University, Mathematics and Statistics
* {{springer|title=Generating function|id=p/g043900}}
* [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/GeneratingFunctions/ "Generating Functions"] by [[Ed Pegg Jr.]], [[Wolfram Demonstrations Project]], 2007.

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{{DEFAULTSORT:Generating Function}}
[[Category:1730 introductions]]
[[Category:Generating functions| ]]
[[Category:Abraham de Moivre]]

Generating function

2024-01-06T18:54:01Z

Yeetcode: Tweaked grammar here and there.

{{Short description|Formal power series; coefficients encode information about a sequence indexed by natural numbers}}
{{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}
{{Very long|date=July 2022}}

In [[mathematics]], a '''generating function''' is a representation of an [[infinite sequence]] of numbers as the [[coefficient]]s of a [[formal power series]]. Unlike an ordinary series, the ''formal'' [[power series]] is not required to [[Convergent series|converge]]: in fact, the generating function is not actually regarded as a [[Function (mathematics)|function]], and the "variable" remains an [[Indeterminate (variable)|indeterminate]]. Generating functions were first introduced by [[Abraham de Moivre]] in 1730, in order to solve the general linear recurrence problem.<ref>{{cite book |author-link=Donald Knuth |first=Donald E. |last=Knuth |series=[[The Art of Computer Programming]] |volume=1 |title=Fundamental Algorithms |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |year=1997 |chapter=§1.2.9 Generating Functions}}</ref> One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

There are various types of generating functions, including '''ordinary generating functions''', '''exponential generating functions''', '''Lambert series''', '''Bell series''', and '''Dirichlet series'''; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in [[Closed-form expression|closed form]] (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate {{mvar|x}} may involve arithmetic operations, differentiation with respect to {{mvar|x}} and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of {{mvar|x}}. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of {{mvar|x}}, and which has the formal series as its [[series expansion]]; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a [[convergent series]] when a nonzero numeric value is substituted for {{mvar|x}}. Also, not all expressions that are meaningful as functions of {{mvar|x}} are meaningful as expressions designating formal series; for example, negative and fractional powers of {{mvar|x}} are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]. Generating functions are sometimes called '''generating series''',<ref>This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", ''[[Canadian Journal of Mathematics]]'' 3, [https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407 p. 405–411], but its use is rare before the year 2000; since then it appears to be increasing.</ref> in that a series of terms can be said to be the generator of its sequence of term coefficients.

==Definitions==

{{block quote
| text = ''A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.''
| author = [[George Pólya]]
| source = ''[[Mathematics and plausible reasoning]]'' (1954) }}

{{block quote
| text = ''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
| author = [[Herbert Wilf]]
| source = ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)}}

===Ordinary generating function (OGF)===

The ''ordinary generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">G(a_n;x)=\sum_{n=0}^\infty a_n x^n.</math>

When the term ''generating function'' is used without qualification, it is usually taken to mean an ordinary generating function.

If {{math|''a''''n''}} is the [[probability mass function]] of a [[discrete random variable]], then its ordinary generating function is called a [[probability-generating function]].

The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array {{math|''a''''m'',''n''}} (where {{mvar|n}} and {{mvar|m}} are natural numbers) is

<math display="block">G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n} x^m y^n.</math>

===Exponential generating function (EGF)===

The ''exponential generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.</math>

Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{harvnb|Flajolet|Sedgewick|2009|p=95}}</ref>

Another benefit of exponential generating functions is that they are useful in transferring linear [[recurrence relations]] to the realm of [[differential equations]]. For example, take the [[Fibonacci sequence]] {{math|{''fn''}<nowiki/>}} that satisfies the linear recurrence relation {{math|''f''''n''+2 {{=}} ''f''''n''+1 + ''f''''n''}}. The corresponding exponential generating function has the form

<math display="block">\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n</math>

and its derivatives can readily be shown to satisfy the differential equation {{math|EF{{pprime}}(''x'') {{=}} EF{{prime}}(''x'') + EF(''x'')}} as a direct analogue with the recurrence relation above. In this view, the factorial term {{math|''n''!}} is merely a counter-term to normalise the derivative operator acting on {{math|''x''''n''}}.

===Poisson generating function===
The ''Poisson generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{PG}(a_n;x)=\sum _{n=0}^\infty a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).</math>

===Lambert series===
{{main article|Lambert series}}
The ''Lambert series'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.</math>

The Lambert series coefficients in the power series expansions

<math display="block">b_n := [x^n] \operatorname{LG}(a_n;x)</math>

for integers {{math|''n'' ≥ 1}} are related by the [[Divisor sum identities|divisor sum]]

<math display="block">b_n = \sum_{d|n} a_d.</math>

The main article provides several more classical, or at least well-known examples related to special [[arithmetic functions]] in [[number theory]].

In a Lambert series the index {{mvar|n}} starts at 1, not at 0, as the first term would otherwise be undefined.

===Bell series===

The [[Bell series]] of a sequence {{math|''a''''n''}} is an expression in terms of both an indeterminate {{mvar|x}} and a prime {{mvar|p}} and is given by<ref>{{Apostol IANT}} pp.42–43</ref>

<math display="block">\operatorname{BG}_p(a_n;x) = \sum_{n=0}^\infty a_{p^n}x^n.</math>

===Dirichlet series generating functions (DGFs)===

[[Formal Dirichlet series]] are often classified as generating functions, although they are not strictly formal power series. The ''Dirichlet series generating function'' of a sequence {{math|''a''''n''}} is<ref name=W56>{{harvnb|Wilf|1994|p=56}}</ref>

<math display="block">\operatorname{DG}(a_n;s)=\sum _{n=1}^\infty \frac{a_n}{n^s}.</math>

The Dirichlet series generating function is especially useful when {{math|''a''''n''}} is a [[multiplicative function]], in which case it has an [[Euler product]] expression<ref name=W59>{{harvnb|Wilf|1994|p=59}}</ref> in terms of the function's Bell series

<math display="block">\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.</math>

If {{math|''a''''n''}} is a [[Dirichlet character]] then its Dirichlet series generating function is called a [[Dirichlet L-series|Dirichlet {{mvar|L}}-series]]. We also have a relation between the pair of coefficients in the [[Lambert series]] expansions above and their DGFs. Namely, we can prove that

<math display="block">[x^n] \operatorname{LG}(a_n; x) = b_n</math>

if and only if

<math display="block">\operatorname{DG}(a_n;s) \zeta(s) = \operatorname{DG}(b_n;s),</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]].<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |last3=Heath-Brown |first3=D.R |last4=Silverman |first4=J.H. |title=An Introduction to the Theory of Numbers|url=https://archive.org/details/introductiontoth00ghha_922|url-access=limited|publisher=Oxford University Press |page=[https://archive.org/details/introductiontoth00ghha_922/page/n357 339]|edition=6th |isbn=9780199219858 |year=2008}}</ref>

===Polynomial sequence generating functions===

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of [[binomial type]] are generated by

<math display="block">e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n</math>

where {{math|''p''''n''(''x'')}} is a sequence of polynomials and {{math|''f''(''t'')}} is a function of a certain form. [[Sheffer sequence]]s are generated in a similar way. See the main article [[generalized Appell polynomials]] for more information.

== Ordinary generating functions ==

=== Examples of generating functions for simple sequences ===

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the [[Poincaré polynomial]] and others.

A fundamental generating function is that of the constant sequence {{nowrap|1, 1, 1, 1, 1, 1, 1, 1, 1, ...}}, whose ordinary generating function is the [[Geometric_series#Closed-form_formula|geometric series]]

<math display="block">\sum_{n=0}^\infty x^n= \frac{1}{1-x}.</math>

The left-hand side is the [[Maclaurin series]] expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by {{math|1 − ''x''}}, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of {{math|''x''0}} are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the [[multiplicative inverse]] of {{math|1 − ''x''}} in the ring of power series.

Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution {{math|''x'' → ''ax''}} gives the generating function for the [[Geometric progression|geometric sequence]] {{math|1, ''a'', ''a''2, ''a''3, ...}} for any constant {{mvar|a}}:

<math display="block">\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.</math>

(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,

<math display="block">\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.</math>

One can also introduce regular gaps in the sequence by replacing {{mvar|x}} by some power of {{mvar|x}}, so for instance for the sequence {{nowrap|1, 0, 1, 0, 1, 0, 1, 0, ...}} (which skips over {{math|''x'', ''x''3, ''x''5, ...}}) one gets the generating function

<math display="block">\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.</math>

By squaring the initial generating function, or by finding the derivative of both sides with respect to {{mvar|x}} and making a change of running variable {{math|''n'' → ''n'' + 1}}, one sees that the coefficients form the sequence {{nowrap|1, 2, 3, 4, 5, ...}}, so one has

<math display="block">\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},</math>

and the third power has as coefficients the [[triangular number]]s {{nowrap|1, 3, 6, 10, 15, 21, ...}} whose term {{mvar|n}} is the [[binomial coefficient]] {{math|{{pars|s=150%|{{su|p=''n'' + 2|b=2|a=c}}}}}}, so that

<math display="block">\sum_{n=0}^\infty\binom{n+2}2 x^n= \frac{1}{(1-x)^3}.</math>

More generally, for any non-negative integer {{mvar|k}} and non-zero real value {{mvar|a}}, it is true that

<math display="block">\sum_{n=0}^\infty a^n\binom{n+k}k x^n= \frac{1}{(1-ax)^{k+1}}\,.</math>

Since

<math display="block">2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0 = 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2,</math>

one can find the ordinary generating function for the sequence {{nowrap|0, 1, 4, 9, 16, ...}} of [[square number]]s by linear combination of binomial-coefficient generating sequences:

<math display="block">G(n^2;x) = \sum_{n=0}^\infty n^2x^n = \frac{2}{(1-x)^3} - \frac{3}{(1-x)^2} + \frac{1}{1-x} = \frac{x(x+1)}{(1-x)^3}.</math>

We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the [[geometric series]] in the following form:

<math display="block">\begin{align}
G(n^2;x)
& = \sum_{n=0}^\infty n^2x^n \\[4px]
& = \sum_{n=0}^\infty n(n-1) x^n + \sum_{n=0}^\infty n x^n \\[4px]
& = x^2 D^2\left[\frac{1}{1-x}\right] + x D\left[\frac{1}{1-x}\right] \\[4px]
& = \frac{2 x^2}{(1-x)^3} + \frac{x}{(1-x)^2} =\frac{x(x+1)}{(1-x)^3}.
\end{align}</math>

By induction, we can similarly show for positive integers {{math|''m'' ≥ 1}} that<ref>{{cite journal|first1= Michael Z. | last1=Spivey | title=Combinatorial Sums and Finite Differences | year=2007 |journal = Discrete Math. |doi = 10.1016/j.disc.2007.03.052 | volume=307|number=24|pages=3130–3146|mr=2370116|doi-access=free }}</ref><ref>{{cite arXiv|first1=R. J. |last1=Mathar|year=2012|eprint=1207.5845|title=Yet another table of integrals|class=math.CA}} v4 eq. (0.4)</ref>

<math display="block">n^m = \sum_{j=0}^m \begin{Bmatrix} m \\ j \end{Bmatrix} \frac{n!}{(n-j)!}, </math>

where {{math|{{resize|150%|{}}{{su|p=''n''|b=''k''}}{{resize|150%|}<nowiki/>}}}} denote the [[Stirling numbers of the second kind]] and where the generating function

<math display="block">\sum_{n = 0}^\infty \frac{n!}{ (n-j)!} \, z^n = \frac{j! \cdot z^j}{(1-z)^{j+1}},</math>

so that we can form the analogous generating functions over the integral {{mvar|m}}th powers generalizing the result in the square case above. In particular, since we can write

<math display="block">\frac{z^k}{(1-z)^{k+1}} = \sum_{i=0}^k \binom{k}{i} \frac{(-1)^{k-i}}{(1-z)^{i+1}},</math>

we can apply a well-known finite sum identity involving the [[Stirling numbers]] to obtain that<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Table 265 in §6.1}} for finite sum identities involving the Stirling number triangles.</ref>

<math display="block">\sum_{n = 0}^\infty n^m z^n = \sum_{j=0}^m \begin{Bmatrix} m+1 \\ j+1 \end{Bmatrix} \frac{(-1)^{m-j} j!}{(1-z)^{j+1}}. </math>

=== Rational functions ===
{{Main|Linear recursive sequence}}
The ordinary generating function of a sequence can be expressed as a [[rational function]] (the ratio of two finite-degree polynomials) if and only if the sequence is a [[linear recursive sequence]] with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear [[finite difference equation]] with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive [[Binet's formula]] for the [[Fibonacci numbers]] via generating function techniques.

We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate ''quasi-polynomial'' sequences of the form <ref name="GFLECT">{{harvnb|Lando|2003|loc=§2.4}}</ref>

<math display="block">f_n = p_1(n) \rho_1^n + \cdots + p_\ell(n) \rho_\ell^n, </math>

where the reciprocal roots, <math>\rho_i \isin \mathbb{C}</math>, are fixed scalars and where {{math|''p''''i''(''n'')}} is a polynomial in {{mvar|n}} for all {{math|1 ≤ ''i'' ≤ ''ℓ''}}.

In general, [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard products]] of rational functions produce rational generating functions. Similarly, if

<math display="block">F(s, t) := \sum_{m,n \geq 0} f(m, n) w^m z^n</math>

is a bivariate rational generating function, then its corresponding ''diagonal generating function'',

<math display="block">\operatorname{diag}(F) := \sum_{n = 0}^\infty f(n, n) z^n,</math>

is ''algebraic''. For example, if we let<ref>Example from {{cite book |chapter=§6.3 |first1=Richard P. |last1=Stanley |first2=Sergey |last2=Fomin |title=Enumerative Combinatorics: Volume 2 |url=https://books.google.com/books?id=zg5wDqT6T-UC |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-78987-5 |series=Cambridge Studies in Advanced Mathematics |volume=62}}</ref>

<math display="block">F(s, t) := \sum_{i,j \geq 0} \binom{i+j}{i} s^i t^j = \frac{1}{1-s-t}, </math>

then this generating function's diagonal coefficient generating function is given by the well-known OGF formula

<math display="block">\operatorname{diag}(F) = \sum_{n = 0}^\infty \binom{2n}{n} z^n = \frac{1}{\sqrt{1-4z}}. </math>

This result is computed in many ways, including [[Cauchy's integral formula]] or [[contour integration]], taking complex [[residue (complex analysis)|residue]]s, or by direct manipulations of [[formal power series]] in two variables.

=== Operations on generating functions ===

==== Multiplication yields convolution ====
{{Main|Cauchy product}}
Multiplication of ordinary generating functions yields a discrete [[convolution]] (the [[Cauchy product]]) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general [[Euler–Maclaurin formula]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''an''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[Generating function#Convolution (Cauchy products)|section on convolutions]] in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.

==== Shifting sequence indices ====

For integers {{math|''m'' ≥ 1}}, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of {{math|⟨ ''g''''n'' − ''m'' ⟩}} and {{math|⟨ ''g''''n'' + ''m'' ⟩}}, respectively:

<math display="block">\begin{align}
& z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n \\[4px]
& \frac{G(z) - g_0 - g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m} = \sum_{n = 0}^\infty g_{n+m} z^n.
\end{align}</math>

==== Differentiation and integration of generating functions ====

We have the following respective power series expansions for the first derivative of a generating function and its integral:

<math display="block">\begin{align}
G'(z) & = \sum_{n = 0}^\infty (n+1) g_{n+1} z^n \\[4px]
z \cdot G'(z) & = \sum_{n = 0}^\infty n g_{n} z^n \\[4px]
\int_0^z G(t) \, dt & = \sum_{n = 1}^\infty \frac{g_{n-1}}{n} z^n.
\end{align}</math>

The differentiation–multiplication operation of the second identity can be repeated {{mvar|k}} times to multiply the sequence by {{math|''n''''k''}}, but that requires alternating between differentiation and multiplication. If instead doing {{mvar|k}} differentiations in sequence, the effect is to multiply by the {{mvar|k}}th [[falling factorial]]:

<math display="block"> z^k G^{(k)}(z) = \sum_{n = 0}^\infty n^\underline{k} g_n z^n = \sum_{n = 0}^\infty n (n-1) \dotsb (n-k+1) g_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

Using the [[Stirling numbers of the second kind]], that can be turned into another formula for multiplying by <math>n^k</math> as follows (see the main article on [[Generating function transformation#Derivative transformations|generating function transformations]]):

<math display="block"> \sum_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} z^j F^{(j)}(z) = \sum_{n = 0}^\infty n^k f_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the [[Generating function transformation#Derivative transformations|zeta series transformation]] and its generalizations defined as a derivative-based [[generating function transformation|transformation of generating functions]], or alternately termwise by and performing an [[Generating function transformation#Polylogarithm series transformations|integral transformation]] on the sequence generating function. Related operations of performing [[fractional calculus|fractional integration]] on a sequence generating function are discussed [[Generating function transformation#Fractional integrals and derivatives|here]].

==== Enumerating arithmetic progressions of sequences ====
In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}}, {{math|0 ≤ ''b'' < ''a''}}, and {{math|''a''}} and {{math|''b''}} are integers (see the [[generating function transformation|main article on transformations]]). For {{math|''a'' {{=}} 2}}, this is simply the familiar decomposition of a function into [[even and odd functions|even and odd parts]] (i.e., even and odd powers):

<math display="block">\begin{align}
\sum_{n = 0}^\infty f_{2n} z^{2n} &= \frac{F(z) + F(-z)}{2} \\[4px]
\sum_{n = 0}^\infty f_{2n+1} z^{2n+1} &= \frac{F(z) - F(-z)}{2}.
\end{align}</math>

More generally, suppose that {{math|''a'' ≥ 3}} and that {{math|''ωa'' {{=}} exp {{sfrac|2''πi''|''a''}}}} denotes the {{mvar|a}}th [[root of unity|primitive root of unity]]. Then, as an application of the [[discrete Fourier transform]], we have the formula<ref name="TAOCPV1">{{harvnb|Knuth|1997|loc=§1.2.9}}</ref>

<math display="block">\sum_{n = 0}^\infty f_{an+b} z^{an+b} = \frac{1}{a} \sum_{m=0}^{a-1} \omega_a^{-mb} F\left(\omega_a^m z\right).</math>

For integers {{math|''m'' ≥ 1}}, another useful formula providing somewhat ''reversed'' floored arithmetic progressions — effectively repeating each coefficient {{mvar|m}} times — are generated by the identity<ref>Solution to {{harvnb|Graham|Knuth|Patashnik|1994|p=569, exercise 7.36}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{\left\lfloor \frac{n}{m} \right\rfloor} z^n = \frac{1-z^m}{1-z} F(z^m) = \left(1 + z + \cdots + z^{m-2} + z^{m-1}\right) F(z^m).</math>

==={{math|''P''}}-recursive sequences and holonomic generating functions===

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>

<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''ci''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> spanned by the set of all of its derivatives is finite dimensional.

Since we can clear denominators if need be in the previous equation, we may assume that the functions, {{math|''ci''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form

<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''0}} and where the {{math|''ĉi''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.

====Examples====

The functions {{math|''e''''z''}}, {{math|log ''z''}}, {{math|cos ''z''}}, {{math|arcsin ''z''}}, <math>\sqrt{1 + z}</math>, the [[dilogarithm]] function {{math|Li2(''z'')}}, the [[generalized hypergeometric function]]s {{math|''pFq''(...; ...; ''z'')}} and the functions defined by the power series

<math display="block">\sum_{n = 0}^\infty \frac{z^n}{(n!)^2}</math>

and the non-convergent

<math display="block">\sum_{n = 0}^\infty n! \cdot z^n</math>

are all holonomic.

Examples of {{mvar|P}}-recursive sequences with holonomic generating functions include {{math|''f''''n'' ≔ {{sfrac|1|''n'' + 1}} {{pars|s=150%|{{su|p=2''n''|b=''n''|a=c}}}}}} and {{math|''f''''n'' ≔ {{sfrac|2''n''|''n''2 + 1}}}}, where sequences such as <math>\sqrt{n}</math> and {{math|log ''n''}} are ''not'' {{mvar|P}}-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as {{math|tan ''z''}}, {{math|sec ''z''}}, and [[Gamma function|{{math|Γ(''z'')}}]] are ''not'' holonomic functions.

====Software for working with ''{{mvar|P}}''-recursive sequences and holonomic generating functions====

Tools for processing and working with {{mvar|P}}-recursive sequences in ''[[Mathematica]]'' include the software packages provided for non-commercial use on the [https://www.risc.jku.at/research/combinat/software/ RISC Combinatorics Group algorithmic combinatorics software] site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>'''Guess'''</code> package for guessing ''{{mvar|P}}-recurrences'' for arbitrary input sequences (useful for [[experimental mathematics]] and exploration) and the <code>'''Sigma'''</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to {{mvar|P}}-recurrences involving generalized [[harmonic number]]s.<ref>{{cite journal|last1=Schneider|first1=C.|title=Symbolic Summation Assists Combinatorics|journal=Sem. Lothar. Combin.|date=2007|volume=56|pages=1–36 |url=http://www.emis.de/journals/SLC/wpapers/s56schneider.html}}</ref> Other packages listed on this particular RISC site are targeted at working with holonomic ''generating functions'' specifically.


=== Relation to discrete-time Fourier transform ===
{{Main|Discrete-time Fourier transform}}
When the series [[Absolute convergence|converges absolutely]],
<math display="block">G \left ( a_n; e^{-i \omega} \right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math>
is the discrete-time Fourier transform of the sequence {{math|''a''0, ''a''1, ...}}.

=== Asymptotic growth of a sequence ===
In calculus, often the growth rate of the coefficients of a power series can be used to deduce a [[radius of convergence]] for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the [[Asymptotic analysis|asymptotic growth]] of the underlying sequence.

For instance, if an ordinary generating function {{math|''G''(''a''''n''; ''x'')}} that has a finite radius of convergence of {{mvar|r}} can be written as

<math display="block">G(a_n; x) = \frac{A(x) + B(x) \left (1- \frac{x}{r} \right )^{-\beta}}{x^\alpha}</math>

where each of {{math|''A''(''x'')}} and {{math|''B''(''x'')}} is a function that is [[analytic function|analytic]] to a radius of convergence greater than {{mvar|r}} (or is [[Entire function|entire]]), and where {{math|''B''(''r'') ≠ 0}} then

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1}\left(\frac{1}{r}\right)^n \sim \frac{B(r)}{r^{\alpha}} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n = \frac{B(r)}{r^\alpha} \left(\!\!\binom{\beta}{n}\!\!\right)\left(\frac{1}{r}\right)^n\,,</math>
using the [[gamma function]], a [[binomial coefficient]], or a [[multiset coefficient]].

Often this approach can be iterated to generate several terms in an asymptotic series for {{math|''a''''n''}}. In particular,

<math display="block">G\left(a_n - \frac{B(r)}{r^\alpha} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n; x \right) = G(a_n; x) - \frac{B(r)}{r^\alpha} \left(1 - \frac{x}{r}\right)^{-\beta}\,.</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of {{mvar|A}}, {{mvar|B}}, {{mvar|α}}, {{mvar|β}}, and {{mvar|r}} to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is {{math|{{sfrac|''a''''n''|''n''!}}}} that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

==== Asymptotic growth of the sequence of squares ====
As derived above, the ordinary generating function for the sequence of squares is

<math display="block">G(n^2; x) = \frac{x(x+1)}{(1-x)^3}.</math>

With {{math|1=''r'' = 1}}, {{math|1=''α'' = −1}}, {{math|1=''β'' = 3}}, {{math|1=''A''(''x'') = 0}}, and {{math|1=''B''(''x'') = ''x'' + 1}}, we can verify that the squares grow as expected, like the squares:

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left (\frac{1}{r} \right)^n = \frac{1+1}{1^{-1}\,\Gamma(3)}\,n^{3-1} \left(\frac1 1\right)^n = n^2.</math>

==== Asymptotic growth of the Catalan numbers ====
{{Main|Catalan number}}

The ordinary generating function for the [[Catalan number]]s is

<math display="block">G(C_n; x) = \frac{1-\sqrt{1-4x}}{2x}.</math>

With {{math|1=''r'' = {{sfrac|1|4}}}}, {{math|1=''α'' = 1}}, {{math|1=''β'' = −{{sfrac|1|2}}}}, {{math|1=''A''(''x'') = {{sfrac|1|2}}}}, and {{math|1=''B''(''x'') = −{{sfrac|1|2}}}}, we can conclude that, for the Catalan numbers,

<math display="block">C_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left(\frac{1}{r} \right)^n = \frac{-\frac12}{\left(\frac14\right)^1 \Gamma\left(-\frac12\right)} \, n^{-\frac12-1} \left(\frac{1}{\,\frac14\,}\right)^n = \frac{4^n}{n^\frac32 \sqrt\pi}.</math>

=== Bivariate and multivariate generating functions ===
One can define generating functions in several variables for arrays with several indices. These are called '''multivariate generating functions''' or, sometimes, '''super generating functions'''. For two variables, these are often called '''bivariate generating functions'''.

For instance, since {{math|(1 + ''x'')''n''}} is the ordinary generating function for [[binomial coefficients]] for a fixed {{mvar|n}}, one may ask for a bivariate generating function that generates the binomial coefficients {{math|{{pars|s=150%|{{su|p=''n''|b=''k''|a=c}}}}}} for all {{mvar|k}} and {{mvar|n}}. To do this, consider {{math|(1 + ''x'')''n''}} itself as a sequence in {{mvar|n}}, and find the generating function in {{mvar|y}} that has these sequence values as coefficients. Since the generating function for {{math|''a''''n''}} is

<math display="block">\frac{1}{1-ay},</math>

the generating function for the binomial coefficients is:

<math display="block">\sum_{n,k} \binom{n}{k} x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}.</math>

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-order accurate]] power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the [[Jacobi-type continued fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>

<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
& = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''jn'' ≔ [''zn''] ''J''[∞](''z'')}}, in the previous equations correspond to matrix solutions of the equations

<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
\begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
\begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''0 ≡ ''k''0,0 {{=}} 1}}, {{math|''jn'' {{=}} ''k''0,''n''}} for {{math|''n'' ≥ 1}}, {{math|''k''''r'',''s'' {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by

<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''[∞](''z'')}}, expanded by

<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''Ph''(''z'')}} and {{math|''Qh''(''z'')}}, defined recursively by

<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv''h''(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''jn''}}, ''and'' for {{math|''Mh'' ≔ ab2 ⋯ ab''h'' + 1}} if {{math|''h'' ‖ ''M''''h''}} then we have the congruence

<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.

==== Examples ====

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''jn''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

:{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.

==Examples==

Generating functions for the sequence of [[square number]]s {{math|''a''''n'' {{=}} ''n''2}} are:

===Ordinary generating function===
<math display="block">G(n^2;x)=\sum_{n=0}^\infty n^2x^n = \frac{x(x+1)}{(1-x)^3}</math>

===Exponential generating function===
<math display="block">\operatorname{EG}(n^2;x)=\sum _{n=0}^\infty \frac{n^2x^n}{n!}=x(x+1)e^x</math>

===Lambert series===

As an example of a Lambert series identity not given in the [[Lambert series|main article]], we can show that for {{math|{{abs|''x''}}, {{abs|''xq''}} < 1}} we have that <ref>{{cite web|title=Lambert series identity|url=https://mathoverflow.net/q/140418 |website=Math Overflow|date=2017}}</ref>

<math display="block">\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n}, </math>

where we have the special case identity for the generating function of the [[divisor function]], {{math|''d''(''n'') ≡ ''σ''0(''n'')}}, given by

<math display="block">\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}. </math>

===Bell series===
<math display="block">\operatorname{BG}_p\left(n^2;x\right)=\sum_{n=0}^\infty \left(p^{n}\right)^2x^n=\frac{1}{1-p^2x}</math>

===Dirichlet series generating function===
<math display="block">\operatorname{DG}\left(n^2;s\right)=\sum_{n=1}^\infty \frac{n^2}{n^s}=\zeta(s-2),</math>

using the [[Riemann zeta function]].

The sequence {{mvar|ak}} generated by a [[Dirichlet series]] generating function (DGF) corresponding to:

<math display="block">\operatorname{DG}(a_k;s)=\zeta(s)^m</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]], has the ordinary generating function:

<math display="block">\sum_{k=1}^{k=n} a_k x^k = x + \binom{m}{1} \sum_{2 \leq a \leq n} x^{a} + \binom{m}{2}\underset{ab \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty} x^{ab} + \binom{m}{3}\underset{abc \leq n}{\sum_{a = 2}^\infty \sum_{c = 2}^\infty \sum_{b = 2}^\infty} x^{abc} + \binom{m}{4}\underset{abcd \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty \sum_{c = 2}^\infty \sum_{d = 2}^\infty} x^{abcd} + \cdots</math>

===Multivariate generating functions===
Multivariate generating functions arise in practice when calculating the number of [[contingency tables]] of non-negative integers with specified row and column totals. Suppose the table has {{mvar|r}} rows and {{mvar|c}} columns; the row sums are {{math|''t''1, ''t''2 ... ''tr''}} and the column sums are {{math|''s''1, ''s''2 ... ''sc''}}. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal| doi=10.1214/aos/1176343649| last=Good| first=I. J.| title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables| journal=[[Annals of Statistics]]| year=1986| volume=4| issue=6|pages=1159–1189| doi-access=free}}</ref> the number of such tables is the coefficient of

<math display="block">x_1^{t_1}\cdots x_r^{t_r}y_1^{s_1}\cdots y_c^{s_c}</math>

in

<math display="block">\prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.</math>

In the bivariate case, non-polynomial double sum examples of so-termed "''double''" or "''super''" generating functions of the form

<math display="block">G(w, z) := \sum_{m,n \geq 0} g_{m,n} w^m z^n</math>

include the following two-variable generating functions for the [[binomial coefficients]], the [[Stirling numbers]], and the [[Eulerian numbers]]:<ref>See the usage of these terms in {{harvnb|Graham|Knuth|Patashnik|1994|loc=§7.4}} on special sequence generating functions.</ref>

<math display="block">\begin{align}
e^{z+wz} & = \sum_{m,n \geq 0} \binom{n}{m} w^m \frac{z^n}{n!} \\[4px]
e^{w(e^z-1)} & = \sum_{m,n \geq 0} \begin{Bmatrix} n \\ m \end{Bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1}{(1-z)^w} & = \sum_{m,n \geq 0} \begin{bmatrix} n \\ m \end{bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1-w}{e^{(w-1)z}-w} & = \sum_{m,n \geq 0} \left\langle\begin{matrix} n \\ m \end{matrix} \right\rangle w^m \frac{z^n}{n!} \\[4px]
\frac{e^w-e^z}{w e^z-z e^w} &= \sum_{m,n \geq 0} \left\langle\begin{matrix} m+n+1 \\ m \end{matrix} \right\rangle \frac{w^m z^n}{(m+n+1)!}.
\end{align}</math>

==Applications==

===Various techniques: Evaluating sums and tackling other problems with generating functions===

====Example 1: A formula for sums of harmonic numbers====

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when {{math|''sn'' {{=}} Σ{{su|b=''k'' {{=}} 0|p=''n''}} ''ak''}}. We then know that {{math|''S''(''z'') {{=}} {{sfrac|''A''(''z'')|1 − ''z''}}}} for the corresponding ordinary generating functions.

For example, we can manipulate
<math display="block">s_n=\sum_{k=1}^{n} H_{k}\,,</math>
where {{math|''Hk'' {{=}} 1 + {{sfrac|1|2}} + ⋯ + {{sfrac|1|''k''}}}} are the [[harmonic number]]s. Let
<math display="block">H(z) = \sum_{n = 1}^\infty{H_n z^n}</math>
be the ordinary generating function of the harmonic numbers. Then
<math display="block">H(z) = \frac{1}{1-z}\sum_{n = 1}^\infty \frac{z^n}{n}\,,</math>
and thus
<math display="block">S(z) = \sum_{n = 1}^\infty{s_n z^n} = \frac{1}{(1-z)^2}\sum_{n = 1}^\infty \frac{z^n}{n}\,.</math>

Using
<math display="block">\frac{1}{(1-z)^2} = \sum_{n = 0}^\infty (n+1)z^n\,,</math>
[[Generating function#Convolution (Cauchy products)|convolution]] with the numerator yields
<math display="block">s_n = \sum_{k = 1}^{n} \frac{n+1-k}{k} = (n+1)H_n - n\,,</math>
which can also be written as
<math display="block">\sum_{k = 1}^{n}{H_k} = (n+1)(H_{n+1} - 1)\,.</math>

====Example 2: Modified binomial coefficient sums and the binomial transform====

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence {{math|⟨ ''fn'' ⟩}} we define the two sequences of sums
<math display="block">\begin{align}
s_n &:= \sum_{m=0}^n \binom{n}{m} f_m 3^{n-m} \\[4px]
\tilde{s}_n &:= \sum_{m=0}^n \binom{n}{m} (m+1)(m+2)(m+3) f_m 3^{n-m}\,,
\end{align}</math>
for all {{math|''n'' ≥ 0}}, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the [[binomial transform]] to write the generating function for the first sum as
<math display="block">S(z) = \frac{1}{1-3z} F\left(\frac{z}{1-3z}\right). </math>

Since the generating function for the sequence {{math|⟨ (''n'' + 1)(''n'' + 2)(''n'' + 3) ''fn'' ⟩}} is given by
<math display="block">6 F(z) + 18z F'(z) + 9z^2 F''(z) + z^3 F'''(z)</math>
we may write the generating function for the second sum defined above in the form
<math display="block">\tilde{S}(z) = \frac{6}{(1-3z)} F\left(\frac{z}{1-3z}\right)+\frac{18z}{(1-3z)^2} F'\left(\frac{z}{1-3z}\right)+\frac{9z^2}{(1-3z)^3} F''\left(\frac{z}{1-3z}\right)+\frac{z^3}{(1-3z)^4} F'''\left(\frac{z}{1-3z}\right). </math>

In particular, we may write this modified sum generating function in the form of
<math display="block">a(z) \cdot S(z) + b(z) \cdot z S'(z) + c(z) \cdot z^2 S''(z) + d(z) \cdot z^3 S'''(z), </math>
for {{math|''a''(''z'') {{=}} 6(1 − 3''z'')3}}, {{math|''b''(''z'') {{=}} 18(1 − 3''z'')3}}, {{math|''c''(''z'') {{=}} 9(1 − 3''z'')3}}, and {{math|''d''(''z'') {{=}} (1 − 3''z'')3}}, where {{math|(1 − 3''z'')3 {{=}} 1 − 9''z'' + 27''z''2 − 27''z''3}}.

Finally, it follows that we may express the second sums through the first sums in the following form:
<math display="block">\begin{align}
\tilde{s}_n & = [z^n]\left(6(1-3z)^3 \sum_{n = 0}^\infty s_n z^n + 18 (1-3z)^3 \sum_{n = 0}^\infty n s_n z^n + 9 (1-3z)^3 \sum_{n = 0}^\infty n(n-1) s_n z^n + (1-3z)^3 \sum_{n = 0}^\infty n(n-1)(n-2) s_n z^n\right) \\[4px]
& = (n+1)(n+2)(n+3) s_n - 9 n(n+1)(n+2) s_{n-1} + 27 (n-1)n(n+1) s_{n-2} - (n-2)(n-1)n s_{n-3}.
\end{align}</math>

====Example 3: Generating functions for mutually recursive sequences====

In this example, we reformulate a generating function example given in Section 7.3 of ''Concrete Mathematics'' (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted {{math|''Un''}}) to tile a 3-by-{{mvar|n}} rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, {{math|''Vn''}}, be defined as the number of ways to cover a 3-by-{{mvar|n}} rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a [[Closed-form expression|closed form]] formula for {{math|''Un''}} without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series

<math display="block">\begin{align}
U(z) = 1 + 3z^2 + 11 z^4 + 41 z^6 + \cdots, \\
V(z) = z + 4z^3 + 15 z^5 + 56 z^7 + \cdots.
\end{align}</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-{{mvar|n}} rectangle, we are able to express the following mutually dependent, or ''mutually recursive'', recurrence relations for our two sequences when {{math|''n'' ≥ 2}} defined as above where {{math|''U''0 {{=}} 1}}, {{math|''U''1 {{=}} 0}}, {{math|''V''0 {{=}} 0}}, and {{math|''V''1 {{=}} 1}}:
<math display="block">\begin{align}
U_n & = 2 V_{n-1} + U_{n-2} \\
V_n & = U_{n-1} + V_{n-2}.
\end{align}</math>

Since we have that for all integers {{math|''m'' ≥ 0}}, the index-shifted generating functions satisfy{{noteTag|Incidentally, we also have a corresponding formula when {{math|''m'' < 0}} given by
<math display="block">\sum_{n = 0}^\infty g_{n+m} z^n = \frac{G(z) - g_0 -g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m}\,.</math>}}
<math display="block">z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n\,,</math>
we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
<math display="block">\begin{align}
U(z) & = 2z V(z) + z^2 U(z) + 1 \\
V(z) & = z U(z) + z^2 V(z) = \frac{z}{1-z^2} U(z),
\end{align}</math>
which then implies by solving the system of equations (and this is the particular trick to our method here) that
<math display="block">U(z) = \frac{1-z^2}{1-4z^2+z^4} = \frac{1}{3-\sqrt{3}} \cdot \frac{1}{1-\left(2+\sqrt{3}\right) z^2} + \frac{1}{3 + \sqrt{3}} \cdot \frac{1}{1-\left(2-\sqrt{3}\right) z^2}. </math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that {{math|''U''2''n'' + 1 ≡ 0}} and that
<math display="block">U_{2n} = \left\lceil \frac{\left(2+\sqrt{3}\right)^n}{3-\sqrt{3}} \right\rceil\,, </math>
for all integers {{math|''n'' ≥ 0}}. We also note that the same shifted generating function technique applied to the second-order [[recurrence relation|recurrence]] for the [[Fibonacci numbers]] is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on [[rational functions]] given above.

===Convolution (Cauchy products)===

A discrete ''convolution'' of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see [[Cauchy product]]).

#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are ordinary generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow [z^n]C(z) = \sum_{k=0}^{n}{a_k b_{n-k}}</math>
#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are exponential generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow \left[\frac{z^n}{n!}\right]C(z) = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}</math>
#Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block">C(z) = F(z) G(z) H(z) \Leftrightarrow [z^n]C(z) = \sum_{j+k+ l=n} f_j g_k h_ l</math>
#Consider the {{mvar|m}}-fold convolution of a sequence with itself for some positive integer {{math|''m'' ≥ 1}} (see the example below for an application) <math display="block">C(z) = G(z)^m \Leftrightarrow [z^n]C(z) = \sum_{k_1+k_2+\cdots+k_m=n} g_{k_1} g_{k_2} \cdots g_{k_m}</math>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the [[probability generating function]], or ''pgf'', of a random variable {{mvar|Z}} is denoted by {{math|''GZ''(''z'')}}, then we can show that for any two random variables <ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=§8.3}}</ref>
<math display="block">G_{X+Y}(z) = G_X(z) G_Y(z)\,, </math>
if {{mvar|X}} and {{mvar|Y}} are independent. Similarly, the number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
<math display="block">C(z) = \frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}, </math>
and moreover, if we allow the {{mvar|n}} cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the [[partition function (mathematics)|partition function]] generating function expanded by the infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product of
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>

====Example: The generating function for the Catalan numbers====

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the [[Catalan numbers]], {{math|''Cn''}}. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product {{math|''x''0 · ''x''1 ·⋯· ''xn''}} so that the order of multiplication is completely specified. For example, {{math|''C''2 {{=}} 2}} which corresponds to the two expressions {{math|''x''0 · (''x''1 · ''x''2)}} and {{math|(''x''0 · ''x''1) · ''x''2}}. It follows that the sequence satisfies a recurrence relation given by
<math display="block">C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k} + \delta_{n,0} = C_0 C_{n-1} + C_1 C_{n-2} + \cdots + C_{n-1} C_0 + \delta_{n,0}\,,\quad n \geq 0\,, </math>
and so has a corresponding convolved generating function, {{math|''C''(''z'')}}, satisfying
<math display="block">C(z) = z \cdot C(z)^2 + 1\,.</math>

Since {{math|''C''(0) {{=}} 1 ≠ ∞}}, we then arrive at a formula for this generating function given by
<math display="block">C(z) = \frac{1-\sqrt{1-4z}}{2z} = \sum_{n = 0}^\infty \frac{1}{n+1}\binom{2n}{n} z^n\,.</math>

Note that the first equation implicitly defining {{math|''C''(''z'')}} above implies that
<math display="block">C(z) = \frac{1}{1-z \cdot C(z)} \,, </math>
which then leads to another "simple" (of form) continued fraction expansion of this generating function.

====Example: Spanning trees of fans and convolutions of convolutions====

A ''fan of order {{mvar|n}}'' is defined to be a graph on the vertices {{math|{0, 1, ..., ''n''}<nowiki/>}} with {{math|2''n'' − 1}} edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other {{mvar|n}} vertices, and vertex <math>k</math> is connected by a single edge to the next vertex {{math|''k'' + 1}} for all {{math|1 ≤ ''k'' < ''n''}}.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Example 6 in §7.3}} for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.</ref> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A [[spanning tree]] is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees {{math|''fn''}} of a fan of order {{mvar|n}} are possible for each {{math|''n'' ≥ 1}}.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when {{math|''n'' {{=}} 4}}, we have that {{math|''f''4 {{=}} 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 {{=}} 21}}, which is a sum over the {{mvar|m}}-fold convolutions of the sequence {{math|''gn'' {{=}} ''n'' {{=}} [''zn''] {{sfrac|''z''|(1 − ''z'')2}}}} for {{math|''m'' ≔ 1, 2, 3, 4}}. More generally, we may write a formula for this sequence as
<math display="block">f_n = \sum_{m > 0} \sum_{\scriptstyle k_1+k_2+\cdots+k_m=n\atop\scriptstyle k_1, k_2, \ldots,k_m > 0} g_{k_1} g_{k_2} \cdots g_{k_m}\,, </math>
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
<math display="block">F(z) = G(z) + G(z)^2 + G(z)^3 + \cdots = \frac{G(z)}{1-G(z)} = \frac{z}{(1-z)^2-z} = \frac{z}{1-3z+z^2}\,,</math>
from which we are able to extract an exact formula for the sequence by taking the [[partial fraction expansion]] of the last generating function.

===Implicit generating functions and the Lagrange inversion formula===
{{expand section|This section needs to be added to the list of techniques with generating functions|date=April 2017}}

===Introducing a free parameter (snake oil method)===
Sometimes the sum {{math|''sn''}} is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have {{mvar|n}} as limit in the summation. When n does not appear explicitly in the summation, we may consider {{mvar|n}} as a "free" parameter and treat {{math|''sn''}} as a coefficient of {{math|''F''(''z'') {{=}} Σ ''sn'' ''zn''}}, change the order of the summations on {{mvar|n}} and {{mvar|k}}, and try to compute the inner sum.

For example, if we want to compute
<math display="block">s_n = \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\,, \quad m,n \in \mathbb{N}_0\,,</math>
we can treat {{mvar|n}} as a "free" parameter, and set
<math display="block">F(z) = \sum_{n = 0}^\infty{\left( \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\right) }z^n\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">F(z) = \sum_{k = 0}^\infty{\binom{2k}{k}\frac{(-1)^k}{k+1} z^{-k}}\sum_{n = 0}^\infty{\binom{n+k}{m+2k} z^{n+k}}\,.</math>

Now the inner sum is {{math|{{sfrac|''z''''m'' + 2''k''|(1 − ''z'')''m'' + 2''k'' + 1}}}}. Thus
<math display="block">\begin{align} F(z)
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{\frac{1}{k+1}\binom{2k}{k}\left(\frac{-z}{(1-z)^2}\right)^k} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{C_k\left(\frac{-z}{(1-z)^2}\right)^k} &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^2}}}{\frac{-2z}{(1-z)^2}} \\[4px]
&= \frac{-z^{m-1}}{2(1-z)^{m-1}}\left(1-\frac{1+z}{1-z}\right) \\[4px]
&= \frac{z^m}{(1-z)^m} = z\frac{z^{m-1}}{(1-z)^m}\,.
\end{align}</math>

Then we obtain
<math display="block">s_n = \begin{cases}
\displaystyle\binom{n-1}{m-1} & \text{for } m \geq 1 \,, \\ {}
[n = 0] & \text{for } m = 0\,.
\end{cases}</math>

It is instructive to use the same method again for the sum, but this time take {{mvar|m}} as the free parameter instead of {{mvar|n}}. We thus set
<math display="block">G(z) = \sum_{m = 0}^\infty\left( \sum_{k = 0}^\infty \binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1} \right) z^m\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">G(z) = \sum_{k = 0}^\infty \binom{2k}{k}\frac{(-1)^k}{k+1} z^{-2k} \sum_{m = 0}^\infty \binom{n+k}{m+2k} z^{m+2k}\,.</math>

Now the inner sum is {{math|(1 + ''z'')''n'' + ''k''}}. Thus
<math display="block">\begin{align} G(z)
&= (1+z)^n \sum_{k = 0}^\infty \frac{1}{k+1}\binom{2k}{k}\left(\frac{-(1+z)}{z^2}\right)^k \\[4px]
&= (1+z)^n \sum_{k = 0}^\infty C_k \,\left(\frac{-(1+z)}{z^2}\right)^k &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= (1+z)^n \,\frac{1-\sqrt{1+\frac{4(1+z)}{z^2}}}{\frac{-2(1+z)}{z^2}} \\[4px]
&= (1+z)^n \,\frac{z^2-z\sqrt{z^2+4+4z}}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{z^2-z(z+2)}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{-2z}{-2(1+z)} = z(1+z)^{n-1}\,.
\end{align}</math>

Thus we obtain
<math display="block">s_n = \left[z^m\right] z(1+z)^{n-1} = \left[z^{m-1}\right] (1+z)^{n-1} = \binom{n-1}{m-1}\,,</math>
for {{math|''m'' ≥ 1}} as before.

===Generating functions prove congruences===
We say that two generating functions (power series) are congruent modulo {{mvar|m}}, written {{math|''A''(''z'') ≡ ''B''(''z'') (mod ''m'')}} if their coefficients are congruent modulo {{mvar|m}} for all {{math|''n'' ≥ 0}}, i.e., {{math|''an'' ≡ ''bn'' (mod ''m'')}} for all relevant cases of the integers {{mvar|n}} (note that we need not assume that {{mvar|m}} is an integer here—it may very well be polynomial-valued in some indeterminate {{mvar|x}}, for example). If the "simpler" right-hand-side generating function, {{math|''B''(''z'')}}, is a rational function of {{mvar|z}}, then the form of this sequence suggests that the sequence is [[periodic function|eventually periodic]] modulo fixed particular cases of integer-valued {{math|''m'' ≥ 2}}. For example, we can prove that the [[Euler numbers]],
<math display="block">\langle E_n \rangle = \langle 1, 1, 5, 61, 1385, \ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle \pmod{3}\,,</math>
satisfy the following congruence modulo 3:<ref>{{harvnb|Lando|2003|loc=§5}}</ref>
<math display="block">\sum_{n = 0}^\infty E_n z^n = \frac{1-z^2}{1+z^2} \pmod{3}\,. </math>

One useful method of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers {{math|''pk''}}) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by {{mvar|J}}-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's ''Lectures on Generating Functions'' as follows:
{{math theorem | name = Theorem: congruences for series generated by expansions of continued fractions
| math_statement = Suppose that the generating function {{math|''A''(''z'')}} is represented by an infinite [[continued fraction]] of the form
<math display="block">A(z) = \cfrac{1}{1-c_1z - \cfrac{p_1z^2}{1-c_2z - \cfrac{p_2 z^2}{1-c_3z - {\ddots}}}}</math>
and that {{math|''Ap''(''z'')}} denotes the {{mvar|p}}th convergent to this continued fraction expansion defined such that {{math|''an'' {{=}} [''zn''] ''Ap''(''z'')}} for all {{math|0 ≤ ''n'' < 2''p''}}. Then:

# the function {{math|''Ap''(''z'')}} is rational for all {{math|''p'' ≥ 2}} where we assume that one of divisibility criteria of {{math|''p'' {{!}} ''p''1, ''p''1''p''2, ''p''1''p''2''p''3}} is met, that is, {{math|''p'' {{!}} ''p''1''p''2⋯''p''''k''}} for some {{math|''k'' ≥ 1}}; and
# if the integer {{mvar|p}} divides the product {{math|''p''1''p''2⋯''p''''k''}}, then we have {{math|''A''(''z'') ≡ ''Ak''(''z'') (mod ''p'')}}.}}

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the [[Stirling numbers of the first kind]] and for the [[partition function (mathematics)|partition function {{math|''p''(''n'')}}]] which show the versatility of generating functions in tackling problems involving [[integer sequences]].

====The Stirling numbers modulo small integers====

The [[Stirling numbers of the first kind#Congruences|main article]] on the Stirling numbers generated by the finite products
<math display="block">S_n(x) := \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k = x(x+1)(x+2) \cdots (x+n-1)\,,\quad n \geq 1\,, </math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference ''Generatingfunctionology''.
We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<math display="block">S_n(x) = [x(x+1)] \cdot [x(x+1)] \cdots = x^{\left\lceil \frac{n}{2} \right\rceil} (x+1)^{\left\lfloor \frac{n}{2} \right\rfloor}\,, </math>

which implies that the parity of these [[Stirling numbers]] matches that of the binomial coefficient

<math display="block">\begin{bmatrix} n \\ k \end{bmatrix} \equiv \binom{\left\lfloor \frac{n}{2} \right\rfloor}{k - \left\lceil \frac{n}{2} \right\rceil} \pmod{2}\,, </math>

and consequently shows that {{math|{{resize|150%|[}}{{su|p=''n''|b=''k''|a=c}}{{resize|150%|]}}}} is even whenever {{math|''k'' < ⌊ {{sfrac|''n''|2}} ⌋}}.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that
<math display="block">\begin{align}
\begin{bmatrix} n \\ m \end{bmatrix} & \equiv
[x^m] \left(
x^{\left\lceil \frac{n}{3} \right\rceil} (x+1)^{\left\lceil \frac{n-1}{3} \right\rceil}
(x+2)^{\left\lfloor \frac{n}{3} \right\rfloor}
\right) && \pmod{3} \\
& \equiv
\sum_{k=0}^{m} \begin{pmatrix} \left\lceil \frac{n-1}{3} \right\rceil \\ k \end{pmatrix}
\begin{pmatrix} \left\lfloor \frac{n}{3} \right\rfloor \\ m-k - \left\lceil \frac{n}{3} \right\rceil \end{pmatrix} \times
2^{\left\lceil \frac{n}{3} \right\rceil + \left\lfloor \frac{n}{3} \right\rfloor -(m-k)} && \pmod{3}\,.
\end{align}</math>

====Congruences for the partition function====

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that ''the'' [[partition function (number theory)|partition function]] {{math|''p''(''n'')}} is generated by the reciprocal infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product (or {{mvar|z}}-Pochhammer product as the case may be) given by
<math display="block">\begin{align}
\sum_{n = 0}^\infty p(n) z^n & = \frac{1}{\left(1-z\right)\left(1-z^2\right)\left(1-z^3\right) \cdots} \\[4pt]
& = 1 + z + 2z^2 + 3 z^3 + 5z^4 + 7z^5 + 11z^6 + \cdots.
\end{align}</math>

This partition function satisfies many known [[Ramanujan's congruences|congruence properties]], which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<ref>{{harvnb|Hardy|Wright|Heath-Brown|Silverman|2008|loc=§19.12}}</ref>
<math display="block">\begin{align}
p(5m+4) & \equiv 0 \pmod{5} \\
p(7m+5) & \equiv 0 \pmod{7} \\
p(11m+6) & \equiv 0 \pmod{11} \\
p(25m+24) & \equiv 0 \pmod{5^2}\,.
\end{align}</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function
<math display=block>\frac{1}{(1-z)^5} = \sum_{i=0}^\infty \binom{4+i}{4}z^i\,,</math>
all of the coefficients are divisible by 5 except for those which correspond to the powers {{math|1, ''z''5, ''z''10, ...}} and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
<math display="block">\frac{1}{(1-z)^5} \equiv \frac{1}{1-z^5} \pmod{5}\,,</math>
or equivalently
<math display="block"> \frac{1-z^5}{(1-z)^5} \equiv 1 \pmod{5}\,.</math>
It follows that
<math display="block">\frac{\left(1-z^5\right)\left(1-z^{10}\right)\left(1-z^{15}\right) \cdots }{\left((1-z)\left(1-z^2\right)\left(1-z^3\right) \cdots \right)^5} \equiv 1 \pmod{5}\,. </math>

Using the infinite product expansions of
<math display="block">z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(1-z\right)\left(1-z^2\right) \cdots } =
z \cdot \left((1-z)\left(1-z^2\right) \cdots \right)^4 \times \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(\left(1-z\right)\left(1-z^2\right) \cdots \right)^5}\,,</math>
it can be shown that the coefficient of {{math|''z''5''m'' + 5}} in {{math|''z'' · ((1 − ''z'')(1 − ''z''2)⋯)4}} is divisible by 5 for all {{mvar|m}}.<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M.|title=An Introduction to the Theory of Numbers}} p.288, Th.361</ref> Finally, since
<math display="block">\begin{align}
\sum_{n = 1}^\infty p(n-1) z^n & = \frac{z}{(1-z)\left(1-z^2\right) \cdots} \\[6px]
& = z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{(1-z)\left(1-z^2\right) \cdots } \times \left(1+z^5+z^{10}+\cdots\right)\left(1+z^{10}+z^{20}+\cdots\right) \cdots
\end{align}</math>
we may equate the coefficients of {{math|''z''5''m'' + 5}} in the previous equations to prove our desired congruence result, namely that {{math|''p''(5''m'' + 4) ≡ 0 (mod 5)}} for all {{math|''m'' ≥ 0}}.

===Transformations of generating functions===
There are a number of transformations of generating functions that provide other applications (see the [[generating function transformation|main article]]). A transformation of a sequence's ''ordinary generating function'' (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see [[Generating function transformation#Integral Transformations|integral transformations]]) or weighted sums over the higher-order derivatives of these functions (see [[Generating function transformation#Derivative Transformations|derivative transformations]]).

Generating function transformations can come into play when we seek to express a generating function for the sums

<math display="block">s_n := \sum_{m=0}^n \binom{n}{m} C_{n,m} a_m, </math>

in the form of {{math|''S''(''z'') {{=}} ''g''(''z'') ''A''(''f''(''z''))}} involving the original sequence generating function. For example, if the sums are
<math display="block">s_n := \sum_{k = 0}^\infty \binom{n+k}{m+2k} a_k \,</math>
then the generating function for the modified sum expressions is given by<ref>{{harvnb|Graham|Knuth|Patashnik|1994|p=535, exercise 5.71}}</ref>
<math display="block">S(z) = \frac{z^m}{(1-z)^{m+1}} A\left(\frac{z}{(1-z)^2}\right)</math>
(see also the [[binomial transform]] and the [[Stirling transform]]).

There are also integral formulas for converting between a sequence's OGF, {{math|''F''(''z'')}}, and its exponential generating function, or EGF, {{math|''F̂''(''z'')}}, and vice versa given by

<math display="block">\begin{align}
F(z) &= \int_0^\infty \hat{F}(tz) e^{-t} \, dt \,, \\[4px]
\hat{F}(z) &= \frac{1}{2\pi} \int_{-\pi}^\pi F\left(z e^{-i\vartheta}\right) e^{e^{i\vartheta}} \, d\vartheta \,,
\end{align}</math>

provided that these integrals converge for appropriate values of {{mvar|z}}.

===Other applications===
Generating functions are used to:

* Find a [[closed formula]] for a sequence given in a recurrence relation. For example, consider [[Fibonacci number#Generating function|Fibonacci numbers]].
* Find [[recurrence relation]]s for sequences—the form of a generating function may suggest a recurrence formula.
* Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.
* Explore the asymptotic behaviour of sequences.
* Prove identities involving sequences.
* Solve [[enumeration]] problems in [[combinatorics]] and encoding their solutions. [[Rook polynomial]]s are an example of an application in combinatorics.
* Evaluate infinite sums.

==Other generating functions==

===Examples===

Examples of [[polynomial sequence]]s generated by more complex generating functions include:

* [[Appell polynomials]]
* [[Chebyshev polynomials]]
* [[Difference polynomials]]
* [[Generalized Appell polynomials]]
* [[Q-difference polynomial|{{mvar|q}}-difference polynomials]]

Other sequences generated by more complex generating functions:

* Double exponential generating functions. For example: [https://oeis.org/search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search Aitken's Array: Triangle of Numbers]
* Hadamard products of generating functions and diagonal generating functions, and their corresponding [[generating function transformation#Hadamard products and diagonal generating functions|integral transformations]]

===Convolution polynomials===

Knuth's article titled "''Convolution Polynomials''"<ref>{{cite journal|last1=Knuth|first1=D. E.|title=Convolution Polynomials|journal=Mathematica J.|date=1992|volume=2|pages=67–78|arxiv=math/9207221|bibcode=1992math......7221K}}</ref> defines a generalized class of ''convolution polynomial'' sequences by their special generating functions of the form
<math display="block">F(z)^x = \exp\bigl(x \log F(z)\bigr) = \sum_{n = 0}^\infty f_n(x) z^n,</math>
for some analytic function {{mvar|F}} with a power series expansion such that {{math|''F''(0) {{=}} 1}}.

We say that a family of polynomials, {{math|''f''0, ''f''1, ''f''2, ...}}, forms a ''convolution family'' if {{math|[[Degree of a polynomial|deg]] ''fn'' ≤ ''n''}} and if the following convolution condition holds for all {{mvar|x}}, {{mvar|y}} and for all {{math|''n'' ≥ 0}}:
<math display="block">f_n(x+y) = f_n(x) f_0(y) + f_{n-1}(x) f_1(y) + \cdots + f_1(x) f_{n-1}(y) + f_0(x) f_n(y). </math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

* The sequence {{math|''n''! · ''fn''(''x'')}} is of [[binomial type]]
* Special values of the sequence include {{math|''fn''(1) {{=}} [''zn''] ''F''(''z'')}} and {{math|''fn''(0) {{=}} ''δ''''n'',0}}, and
* For arbitrary (fixed) <math>x, y, t \isin \mathbb{C}</math>, these polynomials satisfy convolution formulas of the form
<math display="block">\begin{align}
f_n(x+y) & = \sum_{k=0}^n f_k(x) f_{n-k}(y) \\
f_n(2x) & = \sum_{k=0}^n f_k(x) f_{n-k}(x) \\
xn f_n(x+y) & = (x+y) \sum_{k=0}^n k f_k(x) f_{n-k}(y) \\
\frac{(x+y) f_n(x+y+tn)}{x+y+tn} & = \sum_{k=0}^n \frac{x f_k(x+tk)}{x+tk} \frac{y f_{n-k}(y+t(n-k))}{y+t(n-k)}.
\end{align}</math>

For a fixed non-zero parameter <math>t \isin \mathbb{C}</math>, we have modified generating functions for these convolution polynomial sequences given by
<math display="block">\frac{z F_n(x+tn)}{(x+tn)} = \left[z^n\right] \mathcal{F}_t(z)^x, </math>
where {{math|𝓕''t''(''z'')}} is implicitly defined by a [[functional equation]] of the form {{math|𝓕''t''(''z'') {{=}} ''F''(''x''𝓕''t''(''z'')''t'')}}. Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, {{math|⟨ ''fn''(''x'') ⟩}} and {{math|⟨ ''gn''(''x'') ⟩}}, with respective corresponding generating functions, {{math|''F''(''z'')''x''}} and {{math|''G''(''z'')''x''}}, then for arbitrary {{mvar|t}} we have the identity
<math display="block">\left[z^n\right] \left(G(z) F\left(z G(z)^t\right)\right)^x = \sum_{k=0}^n F_k(x) G_{n-k}(x+tk). </math>

Examples of convolution polynomial sequences include the ''binomial power series'', {{math|𝓑''t''(''z'') {{=}} 1 + ''z''𝓑''t''(''z'')''t''}}, so-termed ''tree polynomials'', the [[Bell numbers]], {{math|''B''(''n'')}}, the [[Laguerre polynomials]], and the [[Stirling polynomial|Stirling convolution polynomials]].

===Tables of special generating functions===

An initial listing of special mathematical series is found [[List of mathematical series|here]]. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of ''Concrete Mathematics'' and in Section 2.5 of Wilf's ''Generatingfunctionology''. Other special generating functions of note include the entries in the next table, which is by no means complete.<ref>See also the ''1031 Generating Functions'' found in {{cite thesis |first=Simon |last=Plouffe |title=Approximations de séries génératrices et quelques conjectures |trans-title=Approximations of generating functions and a few conjectures |year=1992 |type=Masters |publisher=Université du Québec à Montréal |language=fr |arxiv=0911.4975}}</ref>

{{expand section|Lists of special and special sequence generating functions. The next table is a start|date=April 2017}}

:{| class="wikitable"
|-
! Formal power series !! Generating-function formula !! Notes
|-
| <math>\sum_{n = 0}^\infty \binom{m+n}{n} \left(H_{n+m}-H_m\right) z^n</math> || <math>\frac{1}{(1-z)^{m+1}} \ln \frac{1}{1-z}</math> || <math>H_n</math> is a first-order [[harmonic number]]
|-
| <math>\sum_{n = 0}^\infty B_n \frac{z^n}{n!}</math> || <math>\frac{z}{e^z-1}</math> || <math>B_n</math> is a [[Bernoulli number]]
|-
| <math>\sum_{n = 0}^\infty F_{mn} z^n</math> || <math>\frac{F_m z}{1-(F_{m-1}+F_{m+1})z+(-1)^m z^2}</math> || <math>F_n</math> is a [[Fibonacci number]] and <math>m \in \mathbb{Z}^{+}</math>
|-
| <math>\sum_{n = 0}^\infty \left\{\begin{matrix} n \\ m \end{matrix} \right\} z^n</math> || <math>(z^{-1})^{\overline{-m}} = \frac{z^m}{(1-z)(1-2z)\cdots(1-mz)}</math> || <math>x^{\overline{n}}</math> denotes the [[rising factorial]], or [[Pochhammer symbol]] and some integer <math>m \geq 0</math>
|-
| <math>\sum_{n = 0}^\infty \left[\begin{matrix} n \\ m \end{matrix} \right] z^n</math> || <math>z^{\overline{m}} = z(z+1) \cdots (z+m-1)</math>
|-
| <math>\sum_{n = 1}^\infty \frac{(-1)^{n-1}4^n (4^n-2) B_{2n} z^{2n}}{(2n) \cdot (2n)!}</math> || <math>\ln \frac{\tan(z)}{z}</math>
|-
| <math>\sum_{n = 0}^\infty \frac{(1/2)^{\overline{n}} z^{2n}}{(2n+1) \cdot n!}</math> || <math>z^{-1} \arcsin(z)</math>
|-
| <math>\sum_{n = 0}^\infty H_n^{(s)} z^n</math> || <math>\frac{\operatorname{Li}_s(z)}{1-z}</math> || <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]] function and <math>H_n^{(s)}</math> is a generalized [[harmonic number]] for <math>\Re(s) > 1</math>
|-
| <math>\sum_{n = 0}^\infty n^m z^n</math> || <math>\sum_{0 \leq j \leq m} \left\{\begin{matrix} m \\ j \end{matrix} \right\} \frac{j! \cdot z^j}{(1-z)^{j+1}}</math> || <math>\left\{\begin{matrix} n \\ m \end{matrix} \right\}</math> is a [[Stirling number of the second kind]] and where the individual terms in the expansion satisfy <math>\frac{z^i}{(1-z)^{i+1}} = \sum_{k=0}^{i} \binom{i}{k} \frac{(-1)^{k-i}}{(1-z)^{k+1}}</math>
|-
| <math>\sum_{k < n} \binom{n-k}{k} \frac{n}{n-k} z^k</math> || <math>\left(\frac{1+\sqrt{1+4z}}{2}\right)^n + \left(\frac{1-\sqrt{1+4z}}{2}\right)^n</math> ||
|-
| <math>\sum_{n_1, \ldots, n_m \geq 0} \min(n_1, \ldots, n_m) z_1^{n_1} \cdots z_m^{n_m}</math> || <math>\frac{z_1 \cdots z_m}{(1-z_1) \cdots (1-z_m) (1-z_1 \cdots z_m)}</math> || The two-variable case is given by <math>M(w, z) := \sum_{m,n \geq 0} \min(m, n) w^m z^n = \frac{wz}{(1-w)(1-z)(1-wz)}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{s}{n} z^n</math> || <math>(1+z)^s</math> || <math>s \in \mathbb{C}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{n}{k} z^n</math> || <math>\frac{z^k}{(1-z)^{k+1}}</math> || <math>k \in \mathbb{N}</math>
|-
|<math>\sum_{n = 1}^\infty \log{(n)} z^n</math>||<math>\left.-\frac{\partial}{\partial s}\operatorname{{Li}_s(z)}\right|_{s=0}</math>||
|}

== History ==
[[George Pólya]] writes in ''[[Mathematics and plausible reasoning]]'':
<blockquote>''The name "generating function" is due to [[Laplace]]. Yet, without giving it a name, [[Euler]] used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the [[Number theory|Theory of Numbers]].''</blockquote>

==See also==
* [[Moment-generating function]]
* [[Probability-generating function]]
* [[Generating function transformation]]
* [[Stanley's reciprocity theorem]]
* Applications to [[Partition (number theory)]]
* [[Combinatorial principles]]
* [[Cyclic sieving]]
* [[Z-transform]]
* [[Umbral calculus]]

==Notes==
{{noteFoot}}

==References==
{{reflist}}

===Citations===
*{{cite book |first=Martin |last=Aigner |title=A Course in Enumeration |url=https://books.google.com/books?id=pPEJcu93dzAC |date=2007 |publisher=Springer |isbn=978-3-540-39035-0 |series=Graduate Texts in Mathematics |volume=238 }}
* {{cite journal |title=On the foundations of combinatorial theory. VI. The idea of generating function |last1=Doubilet |first1=Peter |last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | last3=Stanley | first3=Richard | author3-link=Richard P. Stanley | journal=Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability |volume=2 |pages=267–318 |year=1972 | zbl=0267.05002 | url=http://projecteuclid.org/euclid.bsmsp/1200514223 }} Reprinted in {{cite book | last=Rota | first=Gian-Carlo | author-link=Gian-Carlo Rota | others=With the collaboration of P. Doubilet, C. Greene, D. Kahaner, [[Andrew Odlyzko|A. Odlyzko]] and [[Richard P. Stanley|R. Stanley]] | title=Finite Operator Calculus | chapter=3. The idea of generating function | pages=83–134 | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 | zbl=0328.05007 }}
* {{cite book | last1 = Flajolet | first1 = Philippe | author-link1 = Philippe Flajolet | last2 = Sedgewick | first2 = Robert | author-link2 = Robert Sedgewick (computer scientist) | title = Analytic Combinatorics | title-link= Analytic Combinatorics | year = 2009 | publisher = Cambridge University Press | isbn = 978-0-521-89806-5 | zbl=1165.05001 }}
* {{cite book | last1 = Goulden | first1 = Ian P. | last2 = Jackson | first2 = David M. | author-link2 = David M. Jackson | title = Combinatorial Enumeration | year = 2004 | publisher = [[Dover Publications]] | isbn = 978-0486435978 }}
* {{cite book |title=[[Concrete Mathematics|Concrete Mathematics. A foundation for computer science]] |edition=2nd |year=1994 |publisher=Addison-Wesley |isbn=0-201-55802-5 |chapter=Chapter 7: Generating Functions |pages=320–380| zbl=0836.00001 |first1 = Ronald L. |last1=Graham |first2 = Donald E. |last2=Knuth |first3=Oren |last3=Patashnik |author-link1=Ronald Graham |author-link2=Donald Knuth |author-link3=Oren Patashnik }}
*{{cite book |first=Sergei K. |last=Lando |title=Lectures on Generating Functions |url=https://books.google.com/books?id=A6_4AwAAQBAJ |date=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3481-7 }}
* {{cite book | last=Wilf | first=Herbert S. | author-link=Herbert Wilf | title=Generatingfunctionology | edition=2nd | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 | url=http://www.math.upenn.edu/%7Ewilf/DownldGF.html }}

==External links==
* [http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf "Introduction To Ordinary Generating Functions"] by Mike Zabrocki, York University, Mathematics and Statistics
* {{springer|title=Generating function|id=p/g043900}}
* [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/GeneratingFunctions/ "Generating Functions"] by [[Ed Pegg Jr.]], [[Wolfram Demonstrations Project]], 2007.

{{Authority control}}

{{DEFAULTSORT:Generating Function}}
[[Category:1730 introductions]]
[[Category:Generating functions| ]]
[[Category:Abraham de Moivre]]

User:Yeetcode/sandbox

2024-01-06T18:11:00Z

Yeetcode: Breaking the generating functions page into two. This sandbox will act as a template for the second page.

{{User sandbox}}

{{About|Applications of generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}

==Applications==

===Various techniques: Evaluating sums and tackling other problems with generating functions===

====Example 1: A formula for sums of harmonic numbers====

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when {{math|''sn'' {{=}} Σ{{su|b=''k'' {{=}} 0|p=''n''}} ''ak''}}. We then know that {{math|''S''(''z'') {{=}} {{sfrac|''A''(''z'')|1 − ''z''}}}} for the corresponding ordinary generating functions.

For example, we can manipulate
<math display="block">s_n=\sum_{k=1}^{n} H_{k}\,,</math>
where {{math|''Hk'' {{=}} 1 + {{sfrac|1|2}} + ⋯ + {{sfrac|1|''k''}}}} are the [[harmonic number]]s. Let
<math display="block">H(z) = \sum_{n = 1}^\infty{H_n z^n}</math>
be the ordinary generating function of the harmonic numbers. Then
<math display="block">H(z) = \frac{1}{1-z}\sum_{n = 1}^\infty \frac{z^n}{n}\,,</math>
and thus
<math display="block">S(z) = \sum_{n = 1}^\infty{s_n z^n} = \frac{1}{(1-z)^2}\sum_{n = 1}^\infty \frac{z^n}{n}\,.</math>

Using
<math display="block">\frac{1}{(1-z)^2} = \sum_{n = 0}^\infty (n+1)z^n\,,</math>
[[Generating function#Convolution (Cauchy products)|convolution]] with the numerator yields
<math display="block">s_n = \sum_{k = 1}^{n} \frac{n+1-k}{k} = (n+1)H_n - n\,,</math>
which can also be written as
<math display="block">\sum_{k = 1}^{n}{H_k} = (n+1)(H_{n+1} - 1)\,.</math>

====Example 2: Modified binomial coefficient sums and the binomial transform====

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence {{math|⟨ ''fn'' ⟩}} we define the two sequences of sums
<math display="block">\begin{align}
s_n &:= \sum_{m=0}^n \binom{n}{m} f_m 3^{n-m} \\[4px]
\tilde{s}_n &:= \sum_{m=0}^n \binom{n}{m} (m+1)(m+2)(m+3) f_m 3^{n-m}\,,
\end{align}</math>
for all {{math|''n'' ≥ 0}}, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the [[binomial transform]] to write the generating function for the first sum as
<math display="block">S(z) = \frac{1}{1-3z} F\left(\frac{z}{1-3z}\right). </math>

Since the generating function for the sequence {{math|⟨ (''n'' + 1)(''n'' + 2)(''n'' + 3) ''fn'' ⟩}} is given by
<math display="block">6 F(z) + 18z F'(z) + 9z^2 F''(z) + z^3 F'''(z)</math>
we may write the generating function for the second sum defined above in the form
<math display="block">\tilde{S}(z) = \frac{6}{(1-3z)} F\left(\frac{z}{1-3z}\right)+\frac{18z}{(1-3z)^2} F'\left(\frac{z}{1-3z}\right)+\frac{9z^2}{(1-3z)^3} F''\left(\frac{z}{1-3z}\right)+\frac{z^3}{(1-3z)^4} F'''\left(\frac{z}{1-3z}\right). </math>

In particular, we may write this modified sum generating function in the form of
<math display="block">a(z) \cdot S(z) + b(z) \cdot z S'(z) + c(z) \cdot z^2 S''(z) + d(z) \cdot z^3 S'''(z), </math>
for {{math|''a''(''z'') {{=}} 6(1 − 3''z'')3}}, {{math|''b''(''z'') {{=}} 18(1 − 3''z'')3}}, {{math|''c''(''z'') {{=}} 9(1 − 3''z'')3}}, and {{math|''d''(''z'') {{=}} (1 − 3''z'')3}}, where {{math|(1 − 3''z'')3 {{=}} 1 − 9''z'' + 27''z''2 − 27''z''3}}.

Finally, it follows that we may express the second sums through the first sums in the following form:
<math display="block">\begin{align}
\tilde{s}_n & = [z^n]\left(6(1-3z)^3 \sum_{n = 0}^\infty s_n z^n + 18 (1-3z)^3 \sum_{n = 0}^\infty n s_n z^n + 9 (1-3z)^3 \sum_{n = 0}^\infty n(n-1) s_n z^n + (1-3z)^3 \sum_{n = 0}^\infty n(n-1)(n-2) s_n z^n\right) \\[4px]
& = (n+1)(n+2)(n+3) s_n - 9 n(n+1)(n+2) s_{n-1} + 27 (n-1)n(n+1) s_{n-2} - (n-2)(n-1)n s_{n-3}.
\end{align}</math>

====Example 3: Generating functions for mutually recursive sequences====

In this example, we reformulate a generating function example given in Section 7.3 of ''Concrete Mathematics'' (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted {{math|''Un''}}) to tile a 3-by-{{mvar|n}} rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, {{math|''Vn''}}, be defined as the number of ways to cover a 3-by-{{mvar|n}} rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a [[Closed-form expression|closed form]] formula for {{math|''Un''}} without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series

<math display="block">\begin{align}
U(z) = 1 + 3z^2 + 11 z^4 + 41 z^6 + \cdots, \\
V(z) = z + 4z^3 + 15 z^5 + 56 z^7 + \cdots.
\end{align}</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-{{mvar|n}} rectangle, we are able to express the following mutually dependent, or ''mutually recursive'', recurrence relations for our two sequences when {{math|''n'' ≥ 2}} defined as above where {{math|''U''0 {{=}} 1}}, {{math|''U''1 {{=}} 0}}, {{math|''V''0 {{=}} 0}}, and {{math|''V''1 {{=}} 1}}:
<math display="block">\begin{align}
U_n & = 2 V_{n-1} + U_{n-2} \\
V_n & = U_{n-1} + V_{n-2}.
\end{align}</math>

Since we have that for all integers {{math|''m'' ≥ 0}}, the index-shifted generating functions satisfy{{noteTag|Incidentally, we also have a corresponding formula when {{math|''m'' < 0}} given by
<math display="block">\sum_{n = 0}^\infty g_{n+m} z^n = \frac{G(z) - g_0 -g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m}\,.</math>}}
<math display="block">z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n\,,</math>
we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
<math display="block">\begin{align}
U(z) & = 2z V(z) + z^2 U(z) + 1 \\
V(z) & = z U(z) + z^2 V(z) = \frac{z}{1-z^2} U(z),
\end{align}</math>
which then implies by solving the system of equations (and this is the particular trick to our method here) that
<math display="block">U(z) = \frac{1-z^2}{1-4z^2+z^4} = \frac{1}{3-\sqrt{3}} \cdot \frac{1}{1-\left(2+\sqrt{3}\right) z^2} + \frac{1}{3 + \sqrt{3}} \cdot \frac{1}{1-\left(2-\sqrt{3}\right) z^2}. </math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that {{math|''U''2''n'' + 1 ≡ 0}} and that
<math display="block">U_{2n} = \left\lceil \frac{\left(2+\sqrt{3}\right)^n}{3-\sqrt{3}} \right\rceil\,, </math>
for all integers {{math|''n'' ≥ 0}}. We also note that the same shifted generating function technique applied to the second-order [[recurrence relation|recurrence]] for the [[Fibonacci numbers]] is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on [[rational functions]] given above.

===Convolution (Cauchy products)===

A discrete ''convolution'' of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see [[Cauchy product]]).

#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are ordinary generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow [z^n]C(z) = \sum_{k=0}^{n}{a_k b_{n-k}}</math>
#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are exponential generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow \left[\frac{z^n}{n!}\right]C(z) = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}</math>
#Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block">C(z) = F(z) G(z) H(z) \Leftrightarrow [z^n]C(z) = \sum_{j+k+ l=n} f_j g_k h_ l</math>
#Consider the {{mvar|m}}-fold convolution of a sequence with itself for some positive integer {{math|''m'' ≥ 1}} (see the example below for an application) <math display="block">C(z) = G(z)^m \Leftrightarrow [z^n]C(z) = \sum_{k_1+k_2+\cdots+k_m=n} g_{k_1} g_{k_2} \cdots g_{k_m}</math>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the [[probability generating function]], or ''pgf'', of a random variable {{mvar|Z}} is denoted by {{math|''GZ''(''z'')}}, then we can show that for any two random variables <ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=§8.3}}</ref>
<math display="block">G_{X+Y}(z) = G_X(z) G_Y(z)\,, </math>
if {{mvar|X}} and {{mvar|Y}} are independent. Similarly, the number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
<math display="block">C(z) = \frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}, </math>
and moreover, if we allow the {{mvar|n}} cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the [[partition function (mathematics)|partition function]] generating function expanded by the infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product of
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>

====Example: The generating function for the Catalan numbers====

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the [[Catalan numbers]], {{math|''Cn''}}. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product {{math|''x''0 · ''x''1 ·⋯· ''xn''}} so that the order of multiplication is completely specified. For example, {{math|''C''2 {{=}} 2}} which corresponds to the two expressions {{math|''x''0 · (''x''1 · ''x''2)}} and {{math|(''x''0 · ''x''1) · ''x''2}}. It follows that the sequence satisfies a recurrence relation given by
<math display="block">C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k} + \delta_{n,0} = C_0 C_{n-1} + C_1 C_{n-2} + \cdots + C_{n-1} C_0 + \delta_{n,0}\,,\quad n \geq 0\,, </math>
and so has a corresponding convolved generating function, {{math|''C''(''z'')}}, satisfying
<math display="block">C(z) = z \cdot C(z)^2 + 1\,.</math>

Since {{math|''C''(0) {{=}} 1 ≠ ∞}}, we then arrive at a formula for this generating function given by
<math display="block">C(z) = \frac{1-\sqrt{1-4z}}{2z} = \sum_{n = 0}^\infty \frac{1}{n+1}\binom{2n}{n} z^n\,.</math>

Note that the first equation implicitly defining {{math|''C''(''z'')}} above implies that
<math display="block">C(z) = \frac{1}{1-z \cdot C(z)} \,, </math>
which then leads to another "simple" (of form) continued fraction expansion of this generating function.

====Example: Spanning trees of fans and convolutions of convolutions====

A ''fan of order {{mvar|n}}'' is defined to be a graph on the vertices {{math|{0, 1, ..., ''n''}<nowiki/>}} with {{math|2''n'' − 1}} edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other {{mvar|n}} vertices, and vertex <math>k</math> is connected by a single edge to the next vertex {{math|''k'' + 1}} for all {{math|1 ≤ ''k'' < ''n''}}.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Example 6 in §7.3}} for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.</ref> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A [[spanning tree]] is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees {{math|''fn''}} of a fan of order {{mvar|n}} are possible for each {{math|''n'' ≥ 1}}.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when {{math|''n'' {{=}} 4}}, we have that {{math|''f''4 {{=}} 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 {{=}} 21}}, which is a sum over the {{mvar|m}}-fold convolutions of the sequence {{math|''gn'' {{=}} ''n'' {{=}} [''zn''] {{sfrac|''z''|(1 − ''z'')2}}}} for {{math|''m'' ≔ 1, 2, 3, 4}}. More generally, we may write a formula for this sequence as
<math display="block">f_n = \sum_{m > 0} \sum_{\scriptstyle k_1+k_2+\cdots+k_m=n\atop\scriptstyle k_1, k_2, \ldots,k_m > 0} g_{k_1} g_{k_2} \cdots g_{k_m}\,, </math>
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
<math display="block">F(z) = G(z) + G(z)^2 + G(z)^3 + \cdots = \frac{G(z)}{1-G(z)} = \frac{z}{(1-z)^2-z} = \frac{z}{1-3z+z^2}\,,</math>
from which we are able to extract an exact formula for the sequence by taking the [[partial fraction expansion]] of the last generating function.

===Implicit generating functions and the Lagrange inversion formula===
{{expand section|This section needs to be added to the list of techniques with generating functions|date=April 2017}}

===Introducing a free parameter (snake oil method)===
Sometimes the sum {{math|''sn''}} is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have {{mvar|n}} as limit in the summation. When n does not appear explicitly in the summation, we may consider {{mvar|n}} as a "free" parameter and treat {{math|''sn''}} as a coefficient of {{math|''F''(''z'') {{=}} Σ ''sn'' ''zn''}}, change the order of the summations on {{mvar|n}} and {{mvar|k}}, and try to compute the inner sum.

For example, if we want to compute
<math display="block">s_n = \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\,, \quad m,n \in \mathbb{N}_0\,,</math>
we can treat {{mvar|n}} as a "free" parameter, and set
<math display="block">F(z) = \sum_{n = 0}^\infty{\left( \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\right) }z^n\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">F(z) = \sum_{k = 0}^\infty{\binom{2k}{k}\frac{(-1)^k}{k+1} z^{-k}}\sum_{n = 0}^\infty{\binom{n+k}{m+2k} z^{n+k}}\,.</math>

Now the inner sum is {{math|{{sfrac|''z''''m'' + 2''k''|(1 − ''z'')''m'' + 2''k'' + 1}}}}. Thus
<math display="block">\begin{align} F(z)
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{\frac{1}{k+1}\binom{2k}{k}\left(\frac{-z}{(1-z)^2}\right)^k} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{C_k\left(\frac{-z}{(1-z)^2}\right)^k} &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^2}}}{\frac{-2z}{(1-z)^2}} \\[4px]
&= \frac{-z^{m-1}}{2(1-z)^{m-1}}\left(1-\frac{1+z}{1-z}\right) \\[4px]
&= \frac{z^m}{(1-z)^m} = z\frac{z^{m-1}}{(1-z)^m}\,.
\end{align}</math>

Then we obtain
<math display="block">s_n = \begin{cases}
\displaystyle\binom{n-1}{m-1} & \text{for } m \geq 1 \,, \\ {}
[n = 0] & \text{for } m = 0\,.
\end{cases}</math>

It is instructive to use the same method again for the sum, but this time take {{mvar|m}} as the free parameter instead of {{mvar|n}}. We thus set
<math display="block">G(z) = \sum_{m = 0}^\infty\left( \sum_{k = 0}^\infty \binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1} \right) z^m\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">G(z) = \sum_{k = 0}^\infty \binom{2k}{k}\frac{(-1)^k}{k+1} z^{-2k} \sum_{m = 0}^\infty \binom{n+k}{m+2k} z^{m+2k}\,.</math>

Now the inner sum is {{math|(1 + ''z'')''n'' + ''k''}}. Thus
<math display="block">\begin{align} G(z)
&= (1+z)^n \sum_{k = 0}^\infty \frac{1}{k+1}\binom{2k}{k}\left(\frac{-(1+z)}{z^2}\right)^k \\[4px]
&= (1+z)^n \sum_{k = 0}^\infty C_k \,\left(\frac{-(1+z)}{z^2}\right)^k &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= (1+z)^n \,\frac{1-\sqrt{1+\frac{4(1+z)}{z^2}}}{\frac{-2(1+z)}{z^2}} \\[4px]
&= (1+z)^n \,\frac{z^2-z\sqrt{z^2+4+4z}}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{z^2-z(z+2)}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{-2z}{-2(1+z)} = z(1+z)^{n-1}\,.
\end{align}</math>

Thus we obtain
<math display="block">s_n = \left[z^m\right] z(1+z)^{n-1} = \left[z^{m-1}\right] (1+z)^{n-1} = \binom{n-1}{m-1}\,,</math>
for {{math|''m'' ≥ 1}} as before.

===Generating functions prove congruences===
We say that two generating functions (power series) are congruent modulo {{mvar|m}}, written {{math|''A''(''z'') ≡ ''B''(''z'') (mod ''m'')}} if their coefficients are congruent modulo {{mvar|m}} for all {{math|''n'' ≥ 0}}, i.e., {{math|''an'' ≡ ''bn'' (mod ''m'')}} for all relevant cases of the integers {{mvar|n}} (note that we need not assume that {{mvar|m}} is an integer here—it may very well be polynomial-valued in some indeterminate {{mvar|x}}, for example). If the "simpler" right-hand-side generating function, {{math|''B''(''z'')}}, is a rational function of {{mvar|z}}, then the form of this sequence suggests that the sequence is [[periodic function|eventually periodic]] modulo fixed particular cases of integer-valued {{math|''m'' ≥ 2}}. For example, we can prove that the [[Euler numbers]],
<math display="block">\langle E_n \rangle = \langle 1, 1, 5, 61, 1385, \ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle \pmod{3}\,,</math>
satisfy the following congruence modulo 3:<ref>{{harvnb|Lando|2003|loc=§5}}</ref>
<math display="block">\sum_{n = 0}^\infty E_n z^n = \frac{1-z^2}{1+z^2} \pmod{3}\,. </math>

One of the most useful, if not downright powerful, methods of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers {{math|''pk''}}) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by {{mvar|J}}-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's ''Lectures on Generating Functions'' as follows:
{{math theorem | name = Theorem: congruences for series generated by expansions of continued fractions
| math_statement = Suppose that the generating function {{math|''A''(''z'')}} is represented by an infinite [[continued fraction]] of the form
<math display="block">A(z) = \cfrac{1}{1-c_1z - \cfrac{p_1z^2}{1-c_2z - \cfrac{p_2 z^2}{1-c_3z - {\ddots}}}}</math>
and that {{math|''Ap''(''z'')}} denotes the {{mvar|p}}th convergent to this continued fraction expansion defined such that {{math|''an'' {{=}} [''zn''] ''Ap''(''z'')}} for all {{math|0 ≤ ''n'' < 2''p''}}. Then:

# the function {{math|''Ap''(''z'')}} is rational for all {{math|''p'' ≥ 2}} where we assume that one of divisibility criteria of {{math|''p'' {{!}} ''p''1, ''p''1''p''2, ''p''1''p''2''p''3}} is met, that is, {{math|''p'' {{!}} ''p''1''p''2⋯''p''''k''}} for some {{math|''k'' ≥ 1}}; and
# if the integer {{mvar|p}} divides the product {{math|''p''1''p''2⋯''p''''k''}}, then we have {{math|''A''(''z'') ≡ ''Ak''(''z'') (mod ''p'')}}.}}

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the [[Stirling numbers of the first kind]] and for the [[partition function (mathematics)|partition function {{math|''p''(''n'')}}]] which show the versatility of generating functions in tackling problems involving [[integer sequences]].

====The Stirling numbers modulo small integers====

The [[Stirling numbers of the first kind#Congruences|main article]] on the Stirling numbers generated by the finite products
<math display="block">S_n(x) := \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k = x(x+1)(x+2) \cdots (x+n-1)\,,\quad n \geq 1\,, </math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference ''Generatingfunctionology''.
We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<math display="block">S_n(x) = [x(x+1)] \cdot [x(x+1)] \cdots = x^{\left\lceil \frac{n}{2} \right\rceil} (x+1)^{\left\lfloor \frac{n}{2} \right\rfloor}\,, </math>

which implies that the parity of these [[Stirling numbers]] matches that of the binomial coefficient

<math display="block">\begin{bmatrix} n \\ k \end{bmatrix} \equiv \binom{\left\lfloor \frac{n}{2} \right\rfloor}{k - \left\lceil \frac{n}{2} \right\rceil} \pmod{2}\,, </math>

and consequently shows that {{math|{{resize|150%|[}}{{su|p=''n''|b=''k''|a=c}}{{resize|150%|]}}}} is even whenever {{math|''k'' < ⌊ {{sfrac|''n''|2}} ⌋}}.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that
<math display="block">\begin{align}
\begin{bmatrix} n \\ m \end{bmatrix} & \equiv
[x^m] \left(
x^{\left\lceil \frac{n}{3} \right\rceil} (x+1)^{\left\lceil \frac{n-1}{3} \right\rceil}
(x+2)^{\left\lfloor \frac{n}{3} \right\rfloor}
\right) && \pmod{3} \\
& \equiv
\sum_{k=0}^{m} \begin{pmatrix} \left\lceil \frac{n-1}{3} \right\rceil \\ k \end{pmatrix}
\begin{pmatrix} \left\lfloor \frac{n}{3} \right\rfloor \\ m-k - \left\lceil \frac{n}{3} \right\rceil \end{pmatrix} \times
2^{\left\lceil \frac{n}{3} \right\rceil + \left\lfloor \frac{n}{3} \right\rfloor -(m-k)} && \pmod{3}\,.
\end{align}</math>

====Congruences for the partition function====

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that ''the'' [[partition function (number theory)|partition function]] {{math|''p''(''n'')}} is generated by the reciprocal infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product (or {{mvar|z}}-Pochhammer product as the case may be) given by
<math display="block">\begin{align}
\sum_{n = 0}^\infty p(n) z^n & = \frac{1}{\left(1-z\right)\left(1-z^2\right)\left(1-z^3\right) \cdots} \\[4pt]
& = 1 + z + 2z^2 + 3 z^3 + 5z^4 + 7z^5 + 11z^6 + \cdots.
\end{align}</math>

This partition function satisfies many known [[Ramanujan's congruences|congruence properties]], which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<ref>{{harvnb|Hardy|Wright|Heath-Brown|Silverman|2008|loc=§19.12}}</ref>
<math display="block">\begin{align}
p(5m+4) & \equiv 0 \pmod{5} \\
p(7m+5) & \equiv 0 \pmod{7} \\
p(11m+6) & \equiv 0 \pmod{11} \\
p(25m+24) & \equiv 0 \pmod{5^2}\,.
\end{align}</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function
<math display=block>\frac{1}{(1-z)^5} = \sum_{i=0}^\infty \binom{4+i}{4}z^i\,,</math>
all of the coefficients are divisible by 5 except for those which correspond to the powers {{math|1, ''z''5, ''z''10, ...}} and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
<math display="block">\frac{1}{(1-z)^5} \equiv \frac{1}{1-z^5} \pmod{5}\,,</math>
or equivalently
<math display="block"> \frac{1-z^5}{(1-z)^5} \equiv 1 \pmod{5}\,.</math>
It follows that
<math display="block">\frac{\left(1-z^5\right)\left(1-z^{10}\right)\left(1-z^{15}\right) \cdots }{\left((1-z)\left(1-z^2\right)\left(1-z^3\right) \cdots \right)^5} \equiv 1 \pmod{5}\,. </math>

Using the infinite product expansions of
<math display="block">z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(1-z\right)\left(1-z^2\right) \cdots } =
z \cdot \left((1-z)\left(1-z^2\right) \cdots \right)^4 \times \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(\left(1-z\right)\left(1-z^2\right) \cdots \right)^5}\,,</math>
it can be shown that the coefficient of {{math|''z''5''m'' + 5}} in {{math|''z'' · ((1 − ''z'')(1 − ''z''2)⋯)4}} is divisible by 5 for all {{mvar|m}}.<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M.|title=An Introduction to the Theory of Numbers}} p.288, Th.361</ref> Finally, since
<math display="block">\begin{align}
\sum_{n = 1}^\infty p(n-1) z^n & = \frac{z}{(1-z)\left(1-z^2\right) \cdots} \\[6px]
& = z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{(1-z)\left(1-z^2\right) \cdots } \times \left(1+z^5+z^{10}+\cdots\right)\left(1+z^{10}+z^{20}+\cdots\right) \cdots
\end{align}</math>
we may equate the coefficients of {{math|''z''5''m'' + 5}} in the previous equations to prove our desired congruence result, namely that {{math|''p''(5''m'' + 4) ≡ 0 (mod 5)}} for all {{math|''m'' ≥ 0}}.

Andrey Muchnik

2023-12-19T21:24:27Z

Yeetcode: Fixed grammar

{{Multiple issues|
{{Tone|date=January 2023}}
{{Primary sources|date=January 2023}}
}}
{{Infobox scientist
| name = Andrey Muchnik
| image =
| birth_date = {{Birth date|1958|02|24|}}
| death_date = {{Death-date and age|March 18, 2007|February 24, 1958}}
| citizenship = USSR Russia
| alma_mater = [[Moscow State University]]
| doctoral_advisor = [[Alexey Semyonov (mathematician)|Alexei Semyonov]]
| field = [[Mathematical logic]]
| work_institution = [[Academy of Sciences of the Soviet Union|Scientific Council of the Academy of Sciences of the USSR]] Institute of New Technologies
| prizes = [[Kolmogorov Prize]] (2006)
}}

'''Andrey Albertovich Muchnik''' (February 24, 1958 – March 18, 2007) was a Soviet and Russian mathematician known for his contributions to mathematical logic. He was awarded the [[Kolmogorov Prize|A. N. Kolmogorov Prize]] in 2006.

==Biography==
Muchnik was born on February 24, 1958, to [[Albert Muchnik|Albert Abramovich Muchnik]] and Nadezhda Mitrofanovna Ermolaeva in the Soviet Union. Both parents were mathematicians and students of [[Pyotr Novikov|P. S. Novikov]], a Soviet mathematician. His father, Albert Muchnik, solved Post's problem about the existence of a non-trivial enumerable degree of [[Alan Turing|Turing]] reducibility.

He entered [[Moscow State University]], where he began working as a mathematician at the seminar of [[Evgenii Landis]] and [[Yulij Ilyashenko]] for junior students of the [[MSU Faculty of Mechanics and Mathematics|Faculty of Mechanics and Mathematics of Lomonosov Moscow State University]]. In the second year of his education, he published his first work on [[Differential equation|differential equations]] under the guidance of Ilyashenko.

From his third year at the university onwards, he specialized in definability theory at the Department of Mathematical Logic, where [[Alexei Semenov (mathematician)|Alexei Semenov]] was his supervisor. His diploma (1981) was about the solution of the problem posed by [[Michael Rabin]] at the [[International Congress of Mathematicians]] in Nice to eliminate transfinite induction in the proof of Rabin's theorem on the solvability of the monadic theory of infinite trees. Later, Muchnik used his approach to prove<ref>{{Cite journal|last=Semenov|first=A. L.|date=1984|title=Decidability of Monadic Theories|url=https://link.springer.com/chapter/10.1007/BFb0030296|journal=Mathematical Foundations of Computer Science, Praha, Czechoslovakia, September 3–7, 1984. Proceedings. Lecture Notes in Computer Science|series=Lecture Notes in Computer Science|volume=176|language=en|location=Berlin, Heidelberg|publisher=Springer|pages=162–175|doi=10.1007/BFb0030296|isbn=978-3-540-38929-3}}</ref> a generalization of Rabin's theorem announced by Shelah and Stupp. Using the original idea of [[Alfred Tarski]], he introduced in the notion of self-definability to obtain a short and elegant proof of Cobham-Semenov theorem. He earned his Ph.D. in 2001. <ref>{{Cite news|last=Steinhorn|first=Charles|title=Sept 2008 Newsletter|url=https://aslonline.org/files/newsletters/pdfs/Sept2008newsletter.pdf}}</ref>

Subsequently, he worked at the Institute of New Technologies and the Scientific Council of the USSR Academy of Sciences on the problem of cybernetics{{cn|date=December 2023}}. Eventually, he became one of the leaders of the Kolmogorov seminar at [[Moscow State University]]{{cn|date=December 2023}}.

Muchnik also contributed fundamental results in the field of [[algorithmic information theory]]{{cn|date=December 2023}}. Many results obtained by himself and in collaboration with colleagues were published after his death.<ref>{{Cite web|last1=Адян|first1=С. И.|last2=Семёнов|first2=А. Л.|last3=Успенский|first3=В. А.|title=Андрей Альбертович Мучник (некролог), УМН, 62:4(376) (2007), 140–144; Russian Math. Surveys, 62:4 (2007), 775–779|url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=7922&option_lang=rus|access-date=2023-01-02|website=www.mathnet.ru}}</ref>

==Awards==
Andrey was awarded the A.N. Kolmogorov Prize (together with [[Alexey Semyonov (mathematician)|Alexei Semenov]], 2006) - for his achievements in the field of mathematics for the series of works "On the refinement of A.N. Kolmogorov, related to the theory of chance".<ref>{{Cite web|title=Nominal awards and medals|url=https://www.ras.ru/about/awards/awdlist.aspx?awdid=60|website=www.ras.ru}}</ref>

==References==
{{Reflist}}

==External links==
* [https://www.ras.ru/win/db/show_per.asp?P=.id-58421.ln-ru Muchnik, Andrei Albertovich] on the official website of the [[Russian Academy of Sciences]]
* [http://www.mathnet.ru/php/person.phtml?option_lang=rus&personid=9051 Persons: Muchnik Andrey Albertovich]. mathnet.ru. Retrieved 2016-3-15.
* {{cite web|title=С. И. Адян, А. Л. Семёнов, В. А. Успенский, «Андрей Альбертович Мучник (некролог)», УМН, 62:4(376) (2007), 140–144|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=7922&option_lang=rus|accessdate=2016-03-15|publisher=mathnet.ru}}

{{authority control}}

{{DEFAULTSORT:Muchnik, Andrey}}
[[Category:Russian mathematicians]]
[[Category:Soviet mathematicians]]
[[Category:Moscow State University alumni]]
[[Category:Academic staff of Moscow State University]]
[[Category:1958 births]]
[[Category:2007 deaths]]

Generating function

2023-12-19T21:16:16Z

Yeetcode: Removed broken link with no closely related site.

{{Short description|Formal power series; coefficients encode information about a sequence indexed by natural numbers}}
{{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}
{{Very long|date=July 2022}}

In [[mathematics]], a '''generating function''' is a way of encoding an [[infinite sequence]] of numbers ({{math|''a''''n''}}) by treating them as the [[coefficient]]s of a [[formal power series]]. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' [[power series]] is not required to [[Convergent series|converge]]: in fact, the generating function is not actually regarded as a [[Function (mathematics)|function]], and the "variable" remains an [[Indeterminate (variable)|indeterminate]]. Generating functions were first introduced by [[Abraham de Moivre]] in 1730, in order to solve the general linear recurrence problem.<ref>{{cite book |author-link=Donald Knuth |first=Donald E. |last=Knuth |series=[[The Art of Computer Programming]] |volume=1 |title=Fundamental Algorithms |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |year=1997 |chapter=§1.2.9 Generating Functions}}</ref> One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

There are various types of generating functions, including '''ordinary generating functions''', '''exponential generating functions''', '''Lambert series''', '''Bell series''', and '''Dirichlet series'''; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in [[Closed-form expression|closed form]] (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate {{mvar|x}} may involve arithmetic operations, differentiation with respect to {{mvar|x}} and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of {{mvar|x}}. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of {{mvar|x}}, and which has the formal series as its [[series expansion]]; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a [[convergent series]] when a nonzero numeric value is substituted for {{mvar|x}}. Also, not all expressions that are meaningful as functions of {{mvar|x}} are meaningful as expressions designating formal series; for example, negative and fractional powers of {{mvar|x}} are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]. Generating functions are sometimes called '''generating series''',<ref>This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", ''[[Canadian Journal of Mathematics]]'' 3, [https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407 p. 405–411], but its use is rare before the year 2000; since then it appears to be increasing.</ref> in that a series of terms can be said to be the generator of its sequence of term coefficients.

==Definitions==

{{block quote
| text = ''A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.''
| author = [[George Pólya]]
| source = ''[[Mathematics and plausible reasoning]]'' (1954) }}

{{block quote
| text = ''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
| author = [[Herbert Wilf]]
| source = ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)}}

===Ordinary generating function (OGF)===

The ''ordinary generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">G(a_n;x)=\sum_{n=0}^\infty a_n x^n.</math>

When the term ''generating function'' is used without qualification, it is usually taken to mean an ordinary generating function.

If {{math|''a''''n''}} is the [[probability mass function]] of a [[discrete random variable]], then its ordinary generating function is called a [[probability-generating function]].

The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array {{math|''a''''m'',''n''}} (where {{mvar|n}} and {{mvar|m}} are natural numbers) is

<math display="block">G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n} x^m y^n.</math>

===Exponential generating function (EGF)===

The ''exponential generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.</math>

Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{harvnb|Flajolet|Sedgewick|2009|p=95}}</ref>

Another benefit of exponential generating functions is that they are useful in transferring linear [[recurrence relations]] to the realm of [[differential equations]]. For example, take the [[Fibonacci sequence]] {{math|{''fn''}<nowiki/>}} that satisfies the linear recurrence relation {{math|''f''''n''+2 {{=}} ''f''''n''+1 + ''f''''n''}}. The corresponding exponential generating function has the form

<math display="block">\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n</math>

and its derivatives can readily be shown to satisfy the differential equation {{math|EF{{pprime}}(''x'') {{=}} EF{{prime}}(''x'') + EF(''x'')}} as a direct analogue with the recurrence relation above. In this view, the factorial term {{math|''n''!}} is merely a counter-term to normalise the derivative operator acting on {{math|''x''''n''}}.

===Poisson generating function===
The ''Poisson generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{PG}(a_n;x)=\sum _{n=0}^\infty a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).</math>

===Lambert series===
{{main article|Lambert series}}
The ''Lambert series'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.</math>

The Lambert series coefficients in the power series expansions

<math display="block">b_n := [x^n] \operatorname{LG}(a_n;x)</math>

for integers {{math|''n'' ≥ 1}} are related by the divisor sum

<math display="block">b_n = \sum_{d|n} a_d.</math>

The main article provides several more classical, or at least well-known examples related to special [[arithmetic functions]] in [[number theory]].

In a Lambert series the index {{mvar|n}} starts at 1, not at 0, as the first term would otherwise be undefined.

===Bell series===

The [[Bell series]] of a sequence {{math|''a''''n''}} is an expression in terms of both an indeterminate {{mvar|x}} and a prime {{mvar|p}} and is given by<ref>{{Apostol IANT}} pp.42–43</ref>

<math display="block">\operatorname{BG}_p(a_n;x) = \sum_{n=0}^\infty a_{p^n}x^n.</math>

===Dirichlet series generating functions (DGFs)===

[[Formal Dirichlet series]] are often classified as generating functions, although they are not strictly formal power series. The ''Dirichlet series generating function'' of a sequence {{math|''a''''n''}} is<ref name=W56>{{harvnb|Wilf|1994|p=56}}</ref>

<math display="block">\operatorname{DG}(a_n;s)=\sum _{n=1}^\infty \frac{a_n}{n^s}.</math>

The Dirichlet series generating function is especially useful when {{math|''a''''n''}} is a [[multiplicative function]], in which case it has an [[Euler product]] expression<ref name=W59>{{harvnb|Wilf|1994|p=59}}</ref> in terms of the function's Bell series

<math display="block">\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.</math>

If {{math|''a''''n''}} is a [[Dirichlet character]] then its Dirichlet series generating function is called a [[Dirichlet L-series|Dirichlet {{mvar|L}}-series]]. We also have a relation between the pair of coefficients in the [[Lambert series]] expansions above and their DGFs. Namely, we can prove that

<math display="block">[x^n] \operatorname{LG}(a_n; x) = b_n</math>

if and only if

<math display="block">\operatorname{DG}(a_n;s) \zeta(s) = \operatorname{DG}(b_n;s),</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]].<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |last3=Heath-Brown |first3=D.R |last4=Silverman |first4=J.H. |title=An Introduction to the Theory of Numbers|url=https://archive.org/details/introductiontoth00ghha_922|url-access=limited|publisher=Oxford University Press |page=[https://archive.org/details/introductiontoth00ghha_922/page/n357 339]|edition=6th |isbn=9780199219858 |year=2008}}</ref>

===Polynomial sequence generating functions===

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of [[binomial type]] are generated by

<math display="block">e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n</math>

where {{math|''p''''n''(''x'')}} is a sequence of polynomials and {{math|''f''(''t'')}} is a function of a certain form. [[Sheffer sequence]]s are generated in a similar way. See the main article [[generalized Appell polynomials]] for more information.

== Ordinary generating functions ==

=== Examples of generating functions for simple sequences ===

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the [[Poincaré polynomial]] and others.

A fundamental generating function is that of the constant sequence {{nowrap|1, 1, 1, 1, 1, 1, 1, 1, 1, ...}}, whose ordinary generating function is the [[Geometric_series#Closed-form_formula|geometric series]]

<math display="block">\sum_{n=0}^\infty x^n= \frac{1}{1-x}.</math>

The left-hand side is the [[Maclaurin series]] expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by {{math|1 − ''x''}}, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of {{math|''x''0}} are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the [[multiplicative inverse]] of {{math|1 − ''x''}} in the ring of power series.

Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution {{math|''x'' → ''ax''}} gives the generating function for the [[Geometric progression|geometric sequence]] {{math|1, ''a'', ''a''2, ''a''3, ...}} for any constant {{mvar|a}}:

<math display="block">\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.</math>

(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,

<math display="block">\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.</math>

One can also introduce regular gaps in the sequence by replacing {{mvar|x}} by some power of {{mvar|x}}, so for instance for the sequence {{nowrap|1, 0, 1, 0, 1, 0, 1, 0, ...}} (which skips over {{math|''x'', ''x''3, ''x''5, ...}}) one gets the generating function

<math display="block">\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.</math>

By squaring the initial generating function, or by finding the derivative of both sides with respect to {{mvar|x}} and making a change of running variable {{math|''n'' → ''n'' + 1}}, one sees that the coefficients form the sequence {{nowrap|1, 2, 3, 4, 5, ...}}, so one has

<math display="block">\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},</math>

and the third power has as coefficients the [[triangular number]]s {{nowrap|1, 3, 6, 10, 15, 21, ...}} whose term {{mvar|n}} is the [[binomial coefficient]] {{math|{{pars|s=150%|{{su|p=''n'' + 2|b=2|a=c}}}}}}, so that

<math display="block">\sum_{n=0}^\infty\binom{n+2}2 x^n= \frac{1}{(1-x)^3}.</math>

More generally, for any non-negative integer {{mvar|k}} and non-zero real value {{mvar|a}}, it is true that

<math display="block">\sum_{n=0}^\infty a^n\binom{n+k}k x^n= \frac{1}{(1-ax)^{k+1}}\,.</math>

Since

<math display="block">2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0 = 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2,</math>

one can find the ordinary generating function for the sequence {{nowrap|0, 1, 4, 9, 16, ...}} of [[square number]]s by linear combination of binomial-coefficient generating sequences:

<math display="block">G(n^2;x) = \sum_{n=0}^\infty n^2x^n = \frac{2}{(1-x)^3} - \frac{3}{(1-x)^2} + \frac{1}{1-x} = \frac{x(x+1)}{(1-x)^3}.</math>

We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the [[geometric series]] in the following form:

<math display="block">\begin{align}
G(n^2;x)
& = \sum_{n=0}^\infty n^2x^n \\[4px]
& = \sum_{n=0}^\infty n(n-1) x^n + \sum_{n=0}^\infty n x^n \\[4px]
& = x^2 D^2\left[\frac{1}{1-x}\right] + x D\left[\frac{1}{1-x}\right] \\[4px]
& = \frac{2 x^2}{(1-x)^3} + \frac{x}{(1-x)^2} =\frac{x(x+1)}{(1-x)^3}.
\end{align}</math>

By induction, we can similarly show for positive integers {{math|''m'' ≥ 1}} that<ref>{{cite journal|first1= Michael Z. | last1=Spivey | title=Combinatorial Sums and Finite Differences | year=2007 |journal = Discrete Math. |doi = 10.1016/j.disc.2007.03.052 | volume=307|number=24|pages=3130–3146|mr=2370116|doi-access=free }}</ref><ref>{{cite arXiv|first1=R. J. |last1=Mathar|year=2012|eprint=1207.5845|title=Yet another table of integrals|class=math.CA}} v4 eq. (0.4)</ref>

<math display="block">n^m = \sum_{j=0}^m \begin{Bmatrix} m \\ j \end{Bmatrix} \frac{n!}{(n-j)!}, </math>

where {{math|{{resize|150%|{}}{{su|p=''n''|b=''k''}}{{resize|150%|}<nowiki/>}}}} denote the [[Stirling numbers of the second kind]] and where the generating function

<math display="block">\sum_{n = 0}^\infty \frac{n!}{ (n-j)!} \, z^n = \frac{j! \cdot z^j}{(1-z)^{j+1}},</math>

so that we can form the analogous generating functions over the integral {{mvar|m}}th powers generalizing the result in the square case above. In particular, since we can write

<math display="block">\frac{z^k}{(1-z)^{k+1}} = \sum_{i=0}^k \binom{k}{i} \frac{(-1)^{k-i}}{(1-z)^{i+1}},</math>

we can apply a well-known finite sum identity involving the [[Stirling numbers]] to obtain that<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Table 265 in §6.1}} for finite sum identities involving the Stirling number triangles.</ref>

<math display="block">\sum_{n = 0}^\infty n^m z^n = \sum_{j=0}^m \begin{Bmatrix} m+1 \\ j+1 \end{Bmatrix} \frac{(-1)^{m-j} j!}{(1-z)^{j+1}}. </math>

=== Rational functions ===
{{Main|Linear recursive sequence}}
The ordinary generating function of a sequence can be expressed as a [[rational function]] (the ratio of two finite-degree polynomials) if and only if the sequence is a [[linear recursive sequence]] with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear [[finite difference equation]] with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive [[Binet's formula]] for the [[Fibonacci numbers]] via generating function techniques.

We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate ''quasi-polynomial'' sequences of the form <ref name="GFLECT">{{harvnb|Lando|2003|loc=§2.4}}</ref>

<math display="block">f_n = p_1(n) \rho_1^n + \cdots + p_\ell(n) \rho_\ell^n, </math>

where the reciprocal roots, <math>\rho_i \isin \mathbb{C}</math>, are fixed scalars and where {{math|''p''''i''(''n'')}} is a polynomial in {{mvar|n}} for all {{math|1 ≤ ''i'' ≤ ''ℓ''}}.

In general, [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard products]] of rational functions produce rational generating functions. Similarly, if

<math display="block">F(s, t) := \sum_{m,n \geq 0} f(m, n) w^m z^n</math>

is a bivariate rational generating function, then its corresponding ''diagonal generating function'',

<math display="block">\operatorname{diag}(F) := \sum_{n = 0}^\infty f(n, n) z^n,</math>

is ''algebraic''. For example, if we let<ref>Example from {{cite book |chapter=§6.3 |first1=Richard P. |last1=Stanley |first2=Sergey |last2=Fomin |title=Enumerative Combinatorics: Volume 2 |url=https://books.google.com/books?id=zg5wDqT6T-UC |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-78987-5 |series=Cambridge Studies in Advanced Mathematics |volume=62}}</ref>

<math display="block">F(s, t) := \sum_{i,j \geq 0} \binom{i+j}{i} s^i t^j = \frac{1}{1-s-t}, </math>

then this generating function's diagonal coefficient generating function is given by the well-known OGF formula

<math display="block">\operatorname{diag}(F) = \sum_{n = 0}^\infty \binom{2n}{n} z^n = \frac{1}{\sqrt{1-4z}}. </math>

This result is computed in many ways, including [[Cauchy's integral formula]] or [[contour integration]], taking complex [[residue (complex analysis)|residue]]s, or by direct manipulations of [[formal power series]] in two variables.

=== Operations on generating functions ===

==== Multiplication yields convolution ====
{{Main|Cauchy product}}
Multiplication of ordinary generating functions yields a discrete [[convolution]] (the [[Cauchy product]]) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general [[Euler–Maclaurin formula]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''an''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[Generating function#Convolution (Cauchy products)|section on convolutions]] in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.

==== Shifting sequence indices ====

For integers {{math|''m'' ≥ 1}}, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of {{math|⟨ ''g''''n'' − ''m'' ⟩}} and {{math|⟨ ''g''''n'' + ''m'' ⟩}}, respectively:

<math display="block">\begin{align}
& z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n \\[4px]
& \frac{G(z) - g_0 - g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m} = \sum_{n = 0}^\infty g_{n+m} z^n.
\end{align}</math>

==== Differentiation and integration of generating functions ====

We have the following respective power series expansions for the first derivative of a generating function and its integral:

<math display="block">\begin{align}
G'(z) & = \sum_{n = 0}^\infty (n+1) g_{n+1} z^n \\[4px]
z \cdot G'(z) & = \sum_{n = 0}^\infty n g_{n} z^n \\[4px]
\int_0^z G(t) \, dt & = \sum_{n = 1}^\infty \frac{g_{n-1}}{n} z^n.
\end{align}</math>

The differentiation–multiplication operation of the second identity can be repeated {{mvar|k}} times to multiply the sequence by {{math|''n''''k''}}, but that requires alternating between differentiation and multiplication. If instead doing {{mvar|k}} differentiations in sequence, the effect is to multiply by the {{mvar|k}}th [[falling factorial]]:

<math display="block"> z^k G^{(k)}(z) = \sum_{n = 0}^\infty n^\underline{k} g_n z^n = \sum_{n = 0}^\infty n (n-1) \dotsb (n-k+1) g_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

Using the [[Stirling numbers of the second kind]], that can be turned into another formula for multiplying by <math>n^k</math> as follows (see the main article on [[Generating function transformation#Derivative transformations|generating function transformations]]):

<math display="block"> \sum_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} z^j F^{(j)}(z) = \sum_{n = 0}^\infty n^k f_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the [[Generating function transformation#Derivative transformations|zeta series transformation]] and its generalizations defined as a derivative-based [[generating function transformation|transformation of generating functions]], or alternately termwise by and performing an [[Generating function transformation#Polylogarithm series transformations|integral transformation]] on the sequence generating function. Related operations of performing [[fractional calculus|fractional integration]] on a sequence generating function are discussed [[Generating function transformation#Fractional integrals and derivatives|here]].

==== Enumerating arithmetic progressions of sequences ====
In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}}, {{math|0 ≤ ''b'' < ''a''}}, and {{math|''a''}} and {{math|''b''}} are integers (see the [[generating function transformation|main article on transformations]]). For {{math|''a'' {{=}} 2}}, this is simply the familiar decomposition of a function into [[even and odd functions|even and odd parts]] (i.e., even and odd powers):

<math display="block">\begin{align}
\sum_{n = 0}^\infty f_{2n} z^{2n} &= \frac{F(z) + F(-z)}{2} \\[4px]
\sum_{n = 0}^\infty f_{2n+1} z^{2n+1} &= \frac{F(z) - F(-z)}{2}.
\end{align}</math>

More generally, suppose that {{math|''a'' ≥ 3}} and that {{math|''ωa'' {{=}} exp {{sfrac|2''πi''|''a''}}}} denotes the {{mvar|a}}th [[root of unity|primitive root of unity]]. Then, as an application of the [[discrete Fourier transform]], we have the formula<ref name="TAOCPV1">{{harvnb|Knuth|1997|loc=§1.2.9}}</ref>

<math display="block">\sum_{n = 0}^\infty f_{an+b} z^{an+b} = \frac{1}{a} \sum_{m=0}^{a-1} \omega_a^{-mb} F\left(\omega_a^m z\right).</math>

For integers {{math|''m'' ≥ 1}}, another useful formula providing somewhat ''reversed'' floored arithmetic progressions — effectively repeating each coefficient {{mvar|m}} times — are generated by the identity<ref>Solution to {{harvnb|Graham|Knuth|Patashnik|1994|p=569, exercise 7.36}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{\left\lfloor \frac{n}{m} \right\rfloor} z^n = \frac{1-z^m}{1-z} F(z^m) = \left(1 + z + \cdots + z^{m-2} + z^{m-1}\right) F(z^m).</math>

==={{math|''P''}}-recursive sequences and holonomic generating functions===

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>

<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''ci''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> spanned by the set of all of its derivatives is finite dimensional.

Since we can clear denominators if need be in the previous equation, we may assume that the functions, {{math|''ci''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form

<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''0}} and where the {{math|''ĉi''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.

====Examples====

The functions {{math|''e''''z''}}, {{math|log ''z''}}, {{math|cos ''z''}}, {{math|arcsin ''z''}}, <math>\sqrt{1 + z}</math>, the [[dilogarithm]] function {{math|Li2(''z'')}}, the [[generalized hypergeometric function]]s {{math|''pFq''(...; ...; ''z'')}} and the functions defined by the power series

<math display="block">\sum_{n = 0}^\infty \frac{z^n}{(n!)^2}</math>

and the non-convergent

<math display="block">\sum_{n = 0}^\infty n! \cdot z^n</math>

are all holonomic.

Examples of {{mvar|P}}-recursive sequences with holonomic generating functions include {{math|''f''''n'' ≔ {{sfrac|1|''n'' + 1}} {{pars|s=150%|{{su|p=2''n''|b=''n''|a=c}}}}}} and {{math|''f''''n'' ≔ {{sfrac|2''n''|''n''2 + 1}}}}, where sequences such as <math>\sqrt{n}</math> and {{math|log ''n''}} are ''not'' {{mvar|P}}-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as {{math|tan ''z''}}, {{math|sec ''z''}}, and [[Gamma function|{{math|Γ(''z'')}}]] are ''not'' holonomic functions.

====Software for working with ''{{mvar|P}}''-recursive sequences and holonomic generating functions====

Tools for processing and working with {{mvar|P}}-recursive sequences in ''[[Mathematica]]'' include the software packages provided for non-commercial use on the [https://www.risc.jku.at/research/combinat/software/ RISC Combinatorics Group algorithmic combinatorics software] site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>'''Guess'''</code> package for guessing ''{{mvar|P}}-recurrences'' for arbitrary input sequences (useful for [[experimental mathematics]] and exploration) and the <code>'''Sigma'''</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to {{mvar|P}}-recurrences involving generalized [[harmonic number]]s.<ref>{{cite journal|last1=Schneider|first1=C.|title=Symbolic Summation Assists Combinatorics|journal=Sem. Lothar. Combin.|date=2007|volume=56|pages=1–36 |url=http://www.emis.de/journals/SLC/wpapers/s56schneider.html}}</ref> Other packages listed on this particular RISC site are targeted at working with holonomic ''generating functions'' specifically.


=== Relation to discrete-time Fourier transform ===
{{Main|Discrete-time Fourier transform}}
When the series [[Absolute convergence|converges absolutely]],
<math display="block">G \left ( a_n; e^{-i \omega} \right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math>
is the discrete-time Fourier transform of the sequence {{math|''a''0, ''a''1, ...}}.

=== Asymptotic growth of a sequence ===
In calculus, often the growth rate of the coefficients of a power series can be used to deduce a [[radius of convergence]] for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the [[Asymptotic analysis|asymptotic growth]] of the underlying sequence.

For instance, if an ordinary generating function {{math|''G''(''a''''n''; ''x'')}} that has a finite radius of convergence of {{mvar|r}} can be written as

<math display="block">G(a_n; x) = \frac{A(x) + B(x) \left (1- \frac{x}{r} \right )^{-\beta}}{x^\alpha}</math>

where each of {{math|''A''(''x'')}} and {{math|''B''(''x'')}} is a function that is [[analytic function|analytic]] to a radius of convergence greater than {{mvar|r}} (or is [[Entire function|entire]]), and where {{math|''B''(''r'') ≠ 0}} then

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1}\left(\frac{1}{r}\right)^n \sim \frac{B(r)}{r^{\alpha}} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n = \frac{B(r)}{r^\alpha} \left(\!\!\binom{\beta}{n}\!\!\right)\left(\frac{1}{r}\right)^n\,,</math>
using the [[gamma function]], a [[binomial coefficient]], or a [[multiset coefficient]].

Often this approach can be iterated to generate several terms in an asymptotic series for {{math|''a''''n''}}. In particular,

<math display="block">G\left(a_n - \frac{B(r)}{r^\alpha} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n; x \right) = G(a_n; x) - \frac{B(r)}{r^\alpha} \left(1 - \frac{x}{r}\right)^{-\beta}\,.</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of {{mvar|A}}, {{mvar|B}}, {{mvar|α}}, {{mvar|β}}, and {{mvar|r}} to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is {{math|{{sfrac|''a''''n''|''n''!}}}} that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

==== Asymptotic growth of the sequence of squares ====
As derived above, the ordinary generating function for the sequence of squares is

<math display="block">G(n^2; x) = \frac{x(x+1)}{(1-x)^3}.</math>

With {{math|1=''r'' = 1}}, {{math|1=''α'' = −1}}, {{math|1=''β'' = 3}}, {{math|1=''A''(''x'') = 0}}, and {{math|1=''B''(''x'') = ''x'' + 1}}, we can verify that the squares grow as expected, like the squares:

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left (\frac{1}{r} \right)^n = \frac{1+1}{1^{-1}\,\Gamma(3)}\,n^{3-1} \left(\frac1 1\right)^n = n^2.</math>

==== Asymptotic growth of the Catalan numbers ====
{{Main|Catalan number}}

The ordinary generating function for the [[Catalan number]]s is

<math display="block">G(C_n; x) = \frac{1-\sqrt{1-4x}}{2x}.</math>

With {{math|1=''r'' = {{sfrac|1|4}}}}, {{math|1=''α'' = 1}}, {{math|1=''β'' = −{{sfrac|1|2}}}}, {{math|1=''A''(''x'') = {{sfrac|1|2}}}}, and {{math|1=''B''(''x'') = −{{sfrac|1|2}}}}, we can conclude that, for the Catalan numbers,

<math display="block">C_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left(\frac{1}{r} \right)^n = \frac{-\frac12}{\left(\frac14\right)^1 \Gamma\left(-\frac12\right)} \, n^{-\frac12-1} \left(\frac{1}{\,\frac14\,}\right)^n = \frac{4^n}{n^\frac32 \sqrt\pi}.</math>

=== Bivariate and multivariate generating functions ===
One can define generating functions in several variables for arrays with several indices. These are called '''multivariate generating functions''' or, sometimes, '''super generating functions'''. For two variables, these are often called '''bivariate generating functions'''.

For instance, since {{math|(1 + ''x'')''n''}} is the ordinary generating function for [[binomial coefficients]] for a fixed {{mvar|n}}, one may ask for a bivariate generating function that generates the binomial coefficients {{math|{{pars|s=150%|{{su|p=''n''|b=''k''|a=c}}}}}} for all {{mvar|k}} and {{mvar|n}}. To do this, consider {{math|(1 + ''x'')''n''}} itself as a sequence in {{mvar|n}}, and find the generating function in {{mvar|y}} that has these sequence values as coefficients. Since the generating function for {{math|''a''''n''}} is

<math display="block">\frac{1}{1-ay},</math>

the generating function for the binomial coefficients is:

<math display="block">\sum_{n,k} \binom{n}{k} x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}.</math>

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-order accurate]] power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the [[Jacobi-type continued fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>

<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
& = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''jn'' ≔ [''zn''] ''J''[∞](''z'')}}, in the previous equations correspond to matrix solutions of the equations

<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
\begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
\begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''0 ≡ ''k''0,0 {{=}} 1}}, {{math|''jn'' {{=}} ''k''0,''n''}} for {{math|''n'' ≥ 1}}, {{math|''k''''r'',''s'' {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by

<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''[∞](''z'')}}, expanded by

<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''Ph''(''z'')}} and {{math|''Qh''(''z'')}}, defined recursively by

<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv''h''(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''jn''}}, ''and'' for {{math|''Mh'' ≔ ab2 ⋯ ab''h'' + 1}} if {{math|''h'' ‖ ''M''''h''}} then we have the congruence

<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.

==== Examples ====

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''jn''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

:{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.

==Examples==

Generating functions for the sequence of [[square number]]s {{math|''a''''n'' {{=}} ''n''2}} are:

===Ordinary generating function===
<math display="block">G(n^2;x)=\sum_{n=0}^\infty n^2x^n = \frac{x(x+1)}{(1-x)^3}</math>

===Exponential generating function===
<math display="block">\operatorname{EG}(n^2;x)=\sum _{n=0}^\infty \frac{n^2x^n}{n!}=x(x+1)e^x</math>

===Lambert series===

As an example of a Lambert series identity not given in the [[Lambert series|main article]], we can show that for {{math|{{abs|''x''}}, {{abs|''xq''}} < 1}} we have that <ref>{{cite web|title=Lambert series identity|url=https://mathoverflow.net/q/140418 |website=Math Overflow|date=2017}}</ref>

<math display="block">\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n}, </math>

where we have the special case identity for the generating function of the [[divisor function]], {{math|''d''(''n'') ≡ ''σ''0(''n'')}}, given by

<math display="block">\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}. </math>

===Bell series===
<math display="block">\operatorname{BG}_p\left(n^2;x\right)=\sum_{n=0}^\infty \left(p^{n}\right)^2x^n=\frac{1}{1-p^2x}</math>

===Dirichlet series generating function===
<math display="block">\operatorname{DG}\left(n^2;s\right)=\sum_{n=1}^\infty \frac{n^2}{n^s}=\zeta(s-2),</math>

using the [[Riemann zeta function]].

The sequence {{mvar|ak}} generated by a [[Dirichlet series]] generating function (DGF) corresponding to:

<math display="block">\operatorname{DG}(a_k;s)=\zeta(s)^m</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]], has the ordinary generating function:

<math display="block">\sum_{k=1}^{k=n} a_k x^k = x + \binom{m}{1} \sum_{2 \leq a \leq n} x^{a} + \binom{m}{2}\underset{ab \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty} x^{ab} + \binom{m}{3}\underset{abc \leq n}{\sum_{a = 2}^\infty \sum_{c = 2}^\infty \sum_{b = 2}^\infty} x^{abc} + \binom{m}{4}\underset{abcd \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty \sum_{c = 2}^\infty \sum_{d = 2}^\infty} x^{abcd} + \cdots</math>

===Multivariate generating functions===
Multivariate generating functions arise in practice when calculating the number of [[contingency tables]] of non-negative integers with specified row and column totals. Suppose the table has {{mvar|r}} rows and {{mvar|c}} columns; the row sums are {{math|''t''1, ''t''2 ... ''tr''}} and the column sums are {{math|''s''1, ''s''2 ... ''sc''}}. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal| doi=10.1214/aos/1176343649| last=Good| first=I. J.| title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables| journal=[[Annals of Statistics]]| year=1986| volume=4| issue=6|pages=1159–1189| doi-access=free}}</ref> the number of such tables is the coefficient of

<math display="block">x_1^{t_1}\cdots x_r^{t_r}y_1^{s_1}\cdots y_c^{s_c}</math>

in

<math display="block">\prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.</math>

In the bivariate case, non-polynomial double sum examples of so-termed "''double''" or "''super''" generating functions of the form

<math display="block">G(w, z) := \sum_{m,n \geq 0} g_{m,n} w^m z^n</math>

include the following two-variable generating functions for the [[binomial coefficients]], the [[Stirling numbers]], and the [[Eulerian numbers]]:<ref>See the usage of these terms in {{harvnb|Graham|Knuth|Patashnik|1994|loc=§7.4}} on special sequence generating functions.</ref>

<math display="block">\begin{align}
e^{z+wz} & = \sum_{m,n \geq 0} \binom{n}{m} w^m \frac{z^n}{n!} \\[4px]
e^{w(e^z-1)} & = \sum_{m,n \geq 0} \begin{Bmatrix} n \\ m \end{Bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1}{(1-z)^w} & = \sum_{m,n \geq 0} \begin{bmatrix} n \\ m \end{bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1-w}{e^{(w-1)z}-w} & = \sum_{m,n \geq 0} \left\langle\begin{matrix} n \\ m \end{matrix} \right\rangle w^m \frac{z^n}{n!} \\[4px]
\frac{e^w-e^z}{w e^z-z e^w} &= \sum_{m,n \geq 0} \left\langle\begin{matrix} m+n+1 \\ m \end{matrix} \right\rangle \frac{w^m z^n}{(m+n+1)!}.
\end{align}</math>

==Applications==

===Various techniques: Evaluating sums and tackling other problems with generating functions===

====Example 1: A formula for sums of harmonic numbers====

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when {{math|''sn'' {{=}} Σ{{su|b=''k'' {{=}} 0|p=''n''}} ''ak''}}. We then know that {{math|''S''(''z'') {{=}} {{sfrac|''A''(''z'')|1 − ''z''}}}} for the corresponding ordinary generating functions.

For example, we can manipulate
<math display="block">s_n=\sum_{k=1}^{n} H_{k}\,,</math>
where {{math|''Hk'' {{=}} 1 + {{sfrac|1|2}} + ⋯ + {{sfrac|1|''k''}}}} are the [[harmonic number]]s. Let
<math display="block">H(z) = \sum_{n = 1}^\infty{H_n z^n}</math>
be the ordinary generating function of the harmonic numbers. Then
<math display="block">H(z) = \frac{1}{1-z}\sum_{n = 1}^\infty \frac{z^n}{n}\,,</math>
and thus
<math display="block">S(z) = \sum_{n = 1}^\infty{s_n z^n} = \frac{1}{(1-z)^2}\sum_{n = 1}^\infty \frac{z^n}{n}\,.</math>

Using
<math display="block">\frac{1}{(1-z)^2} = \sum_{n = 0}^\infty (n+1)z^n\,,</math>
[[Generating function#Convolution (Cauchy products)|convolution]] with the numerator yields
<math display="block">s_n = \sum_{k = 1}^{n} \frac{n+1-k}{k} = (n+1)H_n - n\,,</math>
which can also be written as
<math display="block">\sum_{k = 1}^{n}{H_k} = (n+1)(H_{n+1} - 1)\,.</math>

====Example 2: Modified binomial coefficient sums and the binomial transform====

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence {{math|⟨ ''fn'' ⟩}} we define the two sequences of sums
<math display="block">\begin{align}
s_n &:= \sum_{m=0}^n \binom{n}{m} f_m 3^{n-m} \\[4px]
\tilde{s}_n &:= \sum_{m=0}^n \binom{n}{m} (m+1)(m+2)(m+3) f_m 3^{n-m}\,,
\end{align}</math>
for all {{math|''n'' ≥ 0}}, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the [[binomial transform]] to write the generating function for the first sum as
<math display="block">S(z) = \frac{1}{1-3z} F\left(\frac{z}{1-3z}\right). </math>

Since the generating function for the sequence {{math|⟨ (''n'' + 1)(''n'' + 2)(''n'' + 3) ''fn'' ⟩}} is given by
<math display="block">6 F(z) + 18z F'(z) + 9z^2 F''(z) + z^3 F'''(z)</math>
we may write the generating function for the second sum defined above in the form
<math display="block">\tilde{S}(z) = \frac{6}{(1-3z)} F\left(\frac{z}{1-3z}\right)+\frac{18z}{(1-3z)^2} F'\left(\frac{z}{1-3z}\right)+\frac{9z^2}{(1-3z)^3} F''\left(\frac{z}{1-3z}\right)+\frac{z^3}{(1-3z)^4} F'''\left(\frac{z}{1-3z}\right). </math>

In particular, we may write this modified sum generating function in the form of
<math display="block">a(z) \cdot S(z) + b(z) \cdot z S'(z) + c(z) \cdot z^2 S''(z) + d(z) \cdot z^3 S'''(z), </math>
for {{math|''a''(''z'') {{=}} 6(1 − 3''z'')3}}, {{math|''b''(''z'') {{=}} 18(1 − 3''z'')3}}, {{math|''c''(''z'') {{=}} 9(1 − 3''z'')3}}, and {{math|''d''(''z'') {{=}} (1 − 3''z'')3}}, where {{math|(1 − 3''z'')3 {{=}} 1 − 9''z'' + 27''z''2 − 27''z''3}}.

Finally, it follows that we may express the second sums through the first sums in the following form:
<math display="block">\begin{align}
\tilde{s}_n & = [z^n]\left(6(1-3z)^3 \sum_{n = 0}^\infty s_n z^n + 18 (1-3z)^3 \sum_{n = 0}^\infty n s_n z^n + 9 (1-3z)^3 \sum_{n = 0}^\infty n(n-1) s_n z^n + (1-3z)^3 \sum_{n = 0}^\infty n(n-1)(n-2) s_n z^n\right) \\[4px]
& = (n+1)(n+2)(n+3) s_n - 9 n(n+1)(n+2) s_{n-1} + 27 (n-1)n(n+1) s_{n-2} - (n-2)(n-1)n s_{n-3}.
\end{align}</math>

====Example 3: Generating functions for mutually recursive sequences====

In this example, we reformulate a generating function example given in Section 7.3 of ''Concrete Mathematics'' (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted {{math|''Un''}}) to tile a 3-by-{{mvar|n}} rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, {{math|''Vn''}}, be defined as the number of ways to cover a 3-by-{{mvar|n}} rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a [[Closed-form expression|closed form]] formula for {{math|''Un''}} without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series

<math display="block">\begin{align}
U(z) = 1 + 3z^2 + 11 z^4 + 41 z^6 + \cdots, \\
V(z) = z + 4z^3 + 15 z^5 + 56 z^7 + \cdots.
\end{align}</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-{{mvar|n}} rectangle, we are able to express the following mutually dependent, or ''mutually recursive'', recurrence relations for our two sequences when {{math|''n'' ≥ 2}} defined as above where {{math|''U''0 {{=}} 1}}, {{math|''U''1 {{=}} 0}}, {{math|''V''0 {{=}} 0}}, and {{math|''V''1 {{=}} 1}}:
<math display="block">\begin{align}
U_n & = 2 V_{n-1} + U_{n-2} \\
V_n & = U_{n-1} + V_{n-2}.
\end{align}</math>

Since we have that for all integers {{math|''m'' ≥ 0}}, the index-shifted generating functions satisfy{{noteTag|Incidentally, we also have a corresponding formula when {{math|''m'' < 0}} given by
<math display="block">\sum_{n = 0}^\infty g_{n+m} z^n = \frac{G(z) - g_0 -g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m}\,.</math>}}
<math display="block">z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n\,,</math>
we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
<math display="block">\begin{align}
U(z) & = 2z V(z) + z^2 U(z) + 1 \\
V(z) & = z U(z) + z^2 V(z) = \frac{z}{1-z^2} U(z),
\end{align}</math>
which then implies by solving the system of equations (and this is the particular trick to our method here) that
<math display="block">U(z) = \frac{1-z^2}{1-4z^2+z^4} = \frac{1}{3-\sqrt{3}} \cdot \frac{1}{1-\left(2+\sqrt{3}\right) z^2} + \frac{1}{3 + \sqrt{3}} \cdot \frac{1}{1-\left(2-\sqrt{3}\right) z^2}. </math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that {{math|''U''2''n'' + 1 ≡ 0}} and that
<math display="block">U_{2n} = \left\lceil \frac{\left(2+\sqrt{3}\right)^n}{3-\sqrt{3}} \right\rceil\,, </math>
for all integers {{math|''n'' ≥ 0}}. We also note that the same shifted generating function technique applied to the second-order [[recurrence relation|recurrence]] for the [[Fibonacci numbers]] is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on [[rational functions]] given above.

===Convolution (Cauchy products)===

A discrete ''convolution'' of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see [[Cauchy product]]).

#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are ordinary generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow [z^n]C(z) = \sum_{k=0}^{n}{a_k b_{n-k}}</math>
#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are exponential generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow \left[\frac{z^n}{n!}\right]C(z) = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}</math>
#Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block">C(z) = F(z) G(z) H(z) \Leftrightarrow [z^n]C(z) = \sum_{j+k+ l=n} f_j g_k h_ l</math>
#Consider the {{mvar|m}}-fold convolution of a sequence with itself for some positive integer {{math|''m'' ≥ 1}} (see the example below for an application) <math display="block">C(z) = G(z)^m \Leftrightarrow [z^n]C(z) = \sum_{k_1+k_2+\cdots+k_m=n} g_{k_1} g_{k_2} \cdots g_{k_m}</math>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the [[probability generating function]], or ''pgf'', of a random variable {{mvar|Z}} is denoted by {{math|''GZ''(''z'')}}, then we can show that for any two random variables <ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=§8.3}}</ref>
<math display="block">G_{X+Y}(z) = G_X(z) G_Y(z)\,, </math>
if {{mvar|X}} and {{mvar|Y}} are independent. Similarly, the number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
<math display="block">C(z) = \frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}, </math>
and moreover, if we allow the {{mvar|n}} cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the [[partition function (mathematics)|partition function]] generating function expanded by the infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product of
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>

====Example: The generating function for the Catalan numbers====

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the [[Catalan numbers]], {{math|''Cn''}}. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product {{math|''x''0 · ''x''1 ·⋯· ''xn''}} so that the order of multiplication is completely specified. For example, {{math|''C''2 {{=}} 2}} which corresponds to the two expressions {{math|''x''0 · (''x''1 · ''x''2)}} and {{math|(''x''0 · ''x''1) · ''x''2}}. It follows that the sequence satisfies a recurrence relation given by
<math display="block">C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k} + \delta_{n,0} = C_0 C_{n-1} + C_1 C_{n-2} + \cdots + C_{n-1} C_0 + \delta_{n,0}\,,\quad n \geq 0\,, </math>
and so has a corresponding convolved generating function, {{math|''C''(''z'')}}, satisfying
<math display="block">C(z) = z \cdot C(z)^2 + 1\,.</math>

Since {{math|''C''(0) {{=}} 1 ≠ ∞}}, we then arrive at a formula for this generating function given by
<math display="block">C(z) = \frac{1-\sqrt{1-4z}}{2z} = \sum_{n = 0}^\infty \frac{1}{n+1}\binom{2n}{n} z^n\,.</math>

Note that the first equation implicitly defining {{math|''C''(''z'')}} above implies that
<math display="block">C(z) = \frac{1}{1-z \cdot C(z)} \,, </math>
which then leads to another "simple" (of form) continued fraction expansion of this generating function.

====Example: Spanning trees of fans and convolutions of convolutions====

A ''fan of order {{mvar|n}}'' is defined to be a graph on the vertices {{math|{0, 1, ..., ''n''}<nowiki/>}} with {{math|2''n'' − 1}} edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other {{mvar|n}} vertices, and vertex <math>k</math> is connected by a single edge to the next vertex {{math|''k'' + 1}} for all {{math|1 ≤ ''k'' < ''n''}}.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Example 6 in §7.3}} for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.</ref> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A [[spanning tree]] is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees {{math|''fn''}} of a fan of order {{mvar|n}} are possible for each {{math|''n'' ≥ 1}}.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when {{math|''n'' {{=}} 4}}, we have that {{math|''f''4 {{=}} 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 {{=}} 21}}, which is a sum over the {{mvar|m}}-fold convolutions of the sequence {{math|''gn'' {{=}} ''n'' {{=}} [''zn''] {{sfrac|''z''|(1 − ''z'')2}}}} for {{math|''m'' ≔ 1, 2, 3, 4}}. More generally, we may write a formula for this sequence as
<math display="block">f_n = \sum_{m > 0} \sum_{\scriptstyle k_1+k_2+\cdots+k_m=n\atop\scriptstyle k_1, k_2, \ldots,k_m > 0} g_{k_1} g_{k_2} \cdots g_{k_m}\,, </math>
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
<math display="block">F(z) = G(z) + G(z)^2 + G(z)^3 + \cdots = \frac{G(z)}{1-G(z)} = \frac{z}{(1-z)^2-z} = \frac{z}{1-3z+z^2}\,,</math>
from which we are able to extract an exact formula for the sequence by taking the [[partial fraction expansion]] of the last generating function.

===Implicit generating functions and the Lagrange inversion formula===
{{expand section|This section needs to be added to the list of techniques with generating functions|date=April 2017}}

===Introducing a free parameter (snake oil method)===
Sometimes the sum {{math|''sn''}} is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have {{mvar|n}} as limit in the summation. When n does not appear explicitly in the summation, we may consider {{mvar|n}} as a "free" parameter and treat {{math|''sn''}} as a coefficient of {{math|''F''(''z'') {{=}} Σ ''sn'' ''zn''}}, change the order of the summations on {{mvar|n}} and {{mvar|k}}, and try to compute the inner sum.

For example, if we want to compute
<math display="block">s_n = \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\,, \quad m,n \in \mathbb{N}_0\,,</math>
we can treat {{mvar|n}} as a "free" parameter, and set
<math display="block">F(z) = \sum_{n = 0}^\infty{\left( \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\right) }z^n\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">F(z) = \sum_{k = 0}^\infty{\binom{2k}{k}\frac{(-1)^k}{k+1} z^{-k}}\sum_{n = 0}^\infty{\binom{n+k}{m+2k} z^{n+k}}\,.</math>

Now the inner sum is {{math|{{sfrac|''z''''m'' + 2''k''|(1 − ''z'')''m'' + 2''k'' + 1}}}}. Thus
<math display="block">\begin{align} F(z)
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{\frac{1}{k+1}\binom{2k}{k}\left(\frac{-z}{(1-z)^2}\right)^k} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{C_k\left(\frac{-z}{(1-z)^2}\right)^k} &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^2}}}{\frac{-2z}{(1-z)^2}} \\[4px]
&= \frac{-z^{m-1}}{2(1-z)^{m-1}}\left(1-\frac{1+z}{1-z}\right) \\[4px]
&= \frac{z^m}{(1-z)^m} = z\frac{z^{m-1}}{(1-z)^m}\,.
\end{align}</math>

Then we obtain
<math display="block">s_n = \begin{cases}
\displaystyle\binom{n-1}{m-1} & \text{for } m \geq 1 \,, \\ {}
[n = 0] & \text{for } m = 0\,.
\end{cases}</math>

It is instructive to use the same method again for the sum, but this time take {{mvar|m}} as the free parameter instead of {{mvar|n}}. We thus set
<math display="block">G(z) = \sum_{m = 0}^\infty\left( \sum_{k = 0}^\infty \binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1} \right) z^m\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">G(z) = \sum_{k = 0}^\infty \binom{2k}{k}\frac{(-1)^k}{k+1} z^{-2k} \sum_{m = 0}^\infty \binom{n+k}{m+2k} z^{m+2k}\,.</math>

Now the inner sum is {{math|(1 + ''z'')''n'' + ''k''}}. Thus
<math display="block">\begin{align} G(z)
&= (1+z)^n \sum_{k = 0}^\infty \frac{1}{k+1}\binom{2k}{k}\left(\frac{-(1+z)}{z^2}\right)^k \\[4px]
&= (1+z)^n \sum_{k = 0}^\infty C_k \,\left(\frac{-(1+z)}{z^2}\right)^k &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= (1+z)^n \,\frac{1-\sqrt{1+\frac{4(1+z)}{z^2}}}{\frac{-2(1+z)}{z^2}} \\[4px]
&= (1+z)^n \,\frac{z^2-z\sqrt{z^2+4+4z}}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{z^2-z(z+2)}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{-2z}{-2(1+z)} = z(1+z)^{n-1}\,.
\end{align}</math>

Thus we obtain
<math display="block">s_n = \left[z^m\right] z(1+z)^{n-1} = \left[z^{m-1}\right] (1+z)^{n-1} = \binom{n-1}{m-1}\,,</math>
for {{math|''m'' ≥ 1}} as before.

===Generating functions prove congruences===
We say that two generating functions (power series) are congruent modulo {{mvar|m}}, written {{math|''A''(''z'') ≡ ''B''(''z'') (mod ''m'')}} if their coefficients are congruent modulo {{mvar|m}} for all {{math|''n'' ≥ 0}}, i.e., {{math|''an'' ≡ ''bn'' (mod ''m'')}} for all relevant cases of the integers {{mvar|n}} (note that we need not assume that {{mvar|m}} is an integer here—it may very well be polynomial-valued in some indeterminate {{mvar|x}}, for example). If the "simpler" right-hand-side generating function, {{math|''B''(''z'')}}, is a rational function of {{mvar|z}}, then the form of this sequence suggests that the sequence is [[periodic function|eventually periodic]] modulo fixed particular cases of integer-valued {{math|''m'' ≥ 2}}. For example, we can prove that the [[Euler numbers]],
<math display="block">\langle E_n \rangle = \langle 1, 1, 5, 61, 1385, \ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle \pmod{3}\,,</math>
satisfy the following congruence modulo 3:<ref>{{harvnb|Lando|2003|loc=§5}}</ref>
<math display="block">\sum_{n = 0}^\infty E_n z^n = \frac{1-z^2}{1+z^2} \pmod{3}\,. </math>

One of the most useful, if not downright powerful, methods of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers {{math|''pk''}}) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by {{mvar|J}}-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's ''Lectures on Generating Functions'' as follows:
{{math theorem | name = Theorem: congruences for series generated by expansions of continued fractions
| math_statement = Suppose that the generating function {{math|''A''(''z'')}} is represented by an infinite [[continued fraction]] of the form
<math display="block">A(z) = \cfrac{1}{1-c_1z - \cfrac{p_1z^2}{1-c_2z - \cfrac{p_2 z^2}{1-c_3z - {\ddots}}}}</math>
and that {{math|''Ap''(''z'')}} denotes the {{mvar|p}}th convergent to this continued fraction expansion defined such that {{math|''an'' {{=}} [''zn''] ''Ap''(''z'')}} for all {{math|0 ≤ ''n'' < 2''p''}}. Then:

# the function {{math|''Ap''(''z'')}} is rational for all {{math|''p'' ≥ 2}} where we assume that one of divisibility criteria of {{math|''p'' {{!}} ''p''1, ''p''1''p''2, ''p''1''p''2''p''3}} is met, that is, {{math|''p'' {{!}} ''p''1''p''2⋯''p''''k''}} for some {{math|''k'' ≥ 1}}; and
# if the integer {{mvar|p}} divides the product {{math|''p''1''p''2⋯''p''''k''}}, then we have {{math|''A''(''z'') ≡ ''Ak''(''z'') (mod ''p'')}}.}}

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the [[Stirling numbers of the first kind]] and for the [[partition function (mathematics)|partition function {{math|''p''(''n'')}}]] which show the versatility of generating functions in tackling problems involving [[integer sequences]].

====The Stirling numbers modulo small integers====

The [[Stirling numbers of the first kind#Congruences|main article]] on the Stirling numbers generated by the finite products
<math display="block">S_n(x) := \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k = x(x+1)(x+2) \cdots (x+n-1)\,,\quad n \geq 1\,, </math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference ''Generatingfunctionology''.
We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<math display="block">S_n(x) = [x(x+1)] \cdot [x(x+1)] \cdots = x^{\left\lceil \frac{n}{2} \right\rceil} (x+1)^{\left\lfloor \frac{n}{2} \right\rfloor}\,, </math>

which implies that the parity of these [[Stirling numbers]] matches that of the binomial coefficient

<math display="block">\begin{bmatrix} n \\ k \end{bmatrix} \equiv \binom{\left\lfloor \frac{n}{2} \right\rfloor}{k - \left\lceil \frac{n}{2} \right\rceil} \pmod{2}\,, </math>

and consequently shows that {{math|{{resize|150%|[}}{{su|p=''n''|b=''k''|a=c}}{{resize|150%|]}}}} is even whenever {{math|''k'' < ⌊ {{sfrac|''n''|2}} ⌋}}.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that
<math display="block">\begin{align}
\begin{bmatrix} n \\ m \end{bmatrix} & \equiv
[x^m] \left(
x^{\left\lceil \frac{n}{3} \right\rceil} (x+1)^{\left\lceil \frac{n-1}{3} \right\rceil}
(x+2)^{\left\lfloor \frac{n}{3} \right\rfloor}
\right) && \pmod{3} \\
& \equiv
\sum_{k=0}^{m} \begin{pmatrix} \left\lceil \frac{n-1}{3} \right\rceil \\ k \end{pmatrix}
\begin{pmatrix} \left\lfloor \frac{n}{3} \right\rfloor \\ m-k - \left\lceil \frac{n}{3} \right\rceil \end{pmatrix} \times
2^{\left\lceil \frac{n}{3} \right\rceil + \left\lfloor \frac{n}{3} \right\rfloor -(m-k)} && \pmod{3}\,.
\end{align}</math>

====Congruences for the partition function====

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that ''the'' [[partition function (number theory)|partition function]] {{math|''p''(''n'')}} is generated by the reciprocal infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product (or {{mvar|z}}-Pochhammer product as the case may be) given by
<math display="block">\begin{align}
\sum_{n = 0}^\infty p(n) z^n & = \frac{1}{\left(1-z\right)\left(1-z^2\right)\left(1-z^3\right) \cdots} \\[4pt]
& = 1 + z + 2z^2 + 3 z^3 + 5z^4 + 7z^5 + 11z^6 + \cdots.
\end{align}</math>

This partition function satisfies many known [[Ramanujan's congruences|congruence properties]], which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<ref>{{harvnb|Hardy|Wright|Heath-Brown|Silverman|2008|loc=§19.12}}</ref>
<math display="block">\begin{align}
p(5m+4) & \equiv 0 \pmod{5} \\
p(7m+5) & \equiv 0 \pmod{7} \\
p(11m+6) & \equiv 0 \pmod{11} \\
p(25m+24) & \equiv 0 \pmod{5^2}\,.
\end{align}</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function
<math display=block>\frac{1}{(1-z)^5} = \sum_{i=0}^\infty \binom{4+i}{4}z^i\,,</math>
all of the coefficients are divisible by 5 except for those which correspond to the powers {{math|1, ''z''5, ''z''10, ...}} and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
<math display="block">\frac{1}{(1-z)^5} \equiv \frac{1}{1-z^5} \pmod{5}\,,</math>
or equivalently
<math display="block"> \frac{1-z^5}{(1-z)^5} \equiv 1 \pmod{5}\,.</math>
It follows that
<math display="block">\frac{\left(1-z^5\right)\left(1-z^{10}\right)\left(1-z^{15}\right) \cdots }{\left((1-z)\left(1-z^2\right)\left(1-z^3\right) \cdots \right)^5} \equiv 1 \pmod{5}\,. </math>

Using the infinite product expansions of
<math display="block">z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(1-z\right)\left(1-z^2\right) \cdots } =
z \cdot \left((1-z)\left(1-z^2\right) \cdots \right)^4 \times \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(\left(1-z\right)\left(1-z^2\right) \cdots \right)^5}\,,</math>
it can be shown that the coefficient of {{math|''z''5''m'' + 5}} in {{math|''z'' · ((1 − ''z'')(1 − ''z''2)⋯)4}} is divisible by 5 for all {{mvar|m}}.<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M.|title=An Introduction to the Theory of Numbers}} p.288, Th.361</ref> Finally, since
<math display="block">\begin{align}
\sum_{n = 1}^\infty p(n-1) z^n & = \frac{z}{(1-z)\left(1-z^2\right) \cdots} \\[6px]
& = z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{(1-z)\left(1-z^2\right) \cdots } \times \left(1+z^5+z^{10}+\cdots\right)\left(1+z^{10}+z^{20}+\cdots\right) \cdots
\end{align}</math>
we may equate the coefficients of {{math|''z''5''m'' + 5}} in the previous equations to prove our desired congruence result, namely that {{math|''p''(5''m'' + 4) ≡ 0 (mod 5)}} for all {{math|''m'' ≥ 0}}.

===Transformations of generating functions===
There are a number of transformations of generating functions that provide other applications (see the [[generating function transformation|main article]]). A transformation of a sequence's ''ordinary generating function'' (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see [[Generating function transformation#Integral Transformations|integral transformations]]) or weighted sums over the higher-order derivatives of these functions (see [[Generating function transformation#Derivative Transformations|derivative transformations]]).

Generating function transformations can come into play when we seek to express a generating function for the sums

<math display="block">s_n := \sum_{m=0}^n \binom{n}{m} C_{n,m} a_m, </math>

in the form of {{math|''S''(''z'') {{=}} ''g''(''z'') ''A''(''f''(''z''))}} involving the original sequence generating function. For example, if the sums are
<math display="block">s_n := \sum_{k = 0}^\infty \binom{n+k}{m+2k} a_k \,</math>
then the generating function for the modified sum expressions is given by<ref>{{harvnb|Graham|Knuth|Patashnik|1994|p=535, exercise 5.71}}</ref>
<math display="block">S(z) = \frac{z^m}{(1-z)^{m+1}} A\left(\frac{z}{(1-z)^2}\right)</math>
(see also the [[binomial transform]] and the [[Stirling transform]]).

There are also integral formulas for converting between a sequence's OGF, {{math|''F''(''z'')}}, and its exponential generating function, or EGF, {{math|''F̂''(''z'')}}, and vice versa given by

<math display="block">\begin{align}
F(z) &= \int_0^\infty \hat{F}(tz) e^{-t} \, dt \,, \\[4px]
\hat{F}(z) &= \frac{1}{2\pi} \int_{-\pi}^\pi F\left(z e^{-i\vartheta}\right) e^{e^{i\vartheta}} \, d\vartheta \,,
\end{align}</math>

provided that these integrals converge for appropriate values of {{mvar|z}}.

===Other applications===
Generating functions are used to:

* Find a [[closed formula]] for a sequence given in a recurrence relation. For example, consider [[Fibonacci number#Generating function|Fibonacci numbers]].
* Find [[recurrence relation]]s for sequences—the form of a generating function may suggest a recurrence formula.
* Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.
* Explore the asymptotic behaviour of sequences.
* Prove identities involving sequences.
* Solve [[enumeration]] problems in [[combinatorics]] and encoding their solutions. [[Rook polynomial]]s are an example of an application in combinatorics.
* Evaluate infinite sums.

==Other generating functions==

===Examples===

Examples of [[polynomial sequence]]s generated by more complex generating functions include:

* [[Appell polynomials]]
* [[Chebyshev polynomials]]
* [[Difference polynomials]]
* [[Generalized Appell polynomials]]
* [[Q-difference polynomial|{{mvar|q}}-difference polynomials]]

Other sequences generated by more complex generating functions:

* Double exponential generating functions. For example: [https://oeis.org/search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search Aitken's Array: Triangle of Numbers]
* Hadamard products of generating functions and diagonal generating functions, and their corresponding [[generating function transformation#Hadamard products and diagonal generating functions|integral transformations]]

===Convolution polynomials===

Knuth's article titled "''Convolution Polynomials''"<ref>{{cite journal|last1=Knuth|first1=D. E.|title=Convolution Polynomials|journal=Mathematica J.|date=1992|volume=2|pages=67–78|arxiv=math/9207221|bibcode=1992math......7221K}}</ref> defines a generalized class of ''convolution polynomial'' sequences by their special generating functions of the form
<math display="block">F(z)^x = \exp\bigl(x \log F(z)\bigr) = \sum_{n = 0}^\infty f_n(x) z^n,</math>
for some analytic function {{mvar|F}} with a power series expansion such that {{math|''F''(0) {{=}} 1}}.

We say that a family of polynomials, {{math|''f''0, ''f''1, ''f''2, ...}}, forms a ''convolution family'' if {{math|[[Degree of a polynomial|deg]] ''fn'' ≤ ''n''}} and if the following convolution condition holds for all {{mvar|x}}, {{mvar|y}} and for all {{math|''n'' ≥ 0}}:
<math display="block">f_n(x+y) = f_n(x) f_0(y) + f_{n-1}(x) f_1(y) + \cdots + f_1(x) f_{n-1}(y) + f_0(x) f_n(y). </math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

* The sequence {{math|''n''! · ''fn''(''x'')}} is of [[binomial type]]
* Special values of the sequence include {{math|''fn''(1) {{=}} [''zn''] ''F''(''z'')}} and {{math|''fn''(0) {{=}} ''δ''''n'',0}}, and
* For arbitrary (fixed) <math>x, y, t \isin \mathbb{C}</math>, these polynomials satisfy convolution formulas of the form
<math display="block">\begin{align}
f_n(x+y) & = \sum_{k=0}^n f_k(x) f_{n-k}(y) \\
f_n(2x) & = \sum_{k=0}^n f_k(x) f_{n-k}(x) \\
xn f_n(x+y) & = (x+y) \sum_{k=0}^n k f_k(x) f_{n-k}(y) \\
\frac{(x+y) f_n(x+y+tn)}{x+y+tn} & = \sum_{k=0}^n \frac{x f_k(x+tk)}{x+tk} \frac{y f_{n-k}(y+t(n-k))}{y+t(n-k)}.
\end{align}</math>

For a fixed non-zero parameter <math>t \isin \mathbb{C}</math>, we have modified generating functions for these convolution polynomial sequences given by
<math display="block">\frac{z F_n(x+tn)}{(x+tn)} = \left[z^n\right] \mathcal{F}_t(z)^x, </math>
where {{math|𝓕''t''(''z'')}} is implicitly defined by a [[functional equation]] of the form {{math|𝓕''t''(''z'') {{=}} ''F''(''x''𝓕''t''(''z'')''t'')}}. Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, {{math|⟨ ''fn''(''x'') ⟩}} and {{math|⟨ ''gn''(''x'') ⟩}}, with respective corresponding generating functions, {{math|''F''(''z'')''x''}} and {{math|''G''(''z'')''x''}}, then for arbitrary {{mvar|t}} we have the identity
<math display="block">\left[z^n\right] \left(G(z) F\left(z G(z)^t\right)\right)^x = \sum_{k=0}^n F_k(x) G_{n-k}(x+tk). </math>

Examples of convolution polynomial sequences include the ''binomial power series'', {{math|𝓑''t''(''z'') {{=}} 1 + ''z''𝓑''t''(''z'')''t''}}, so-termed ''tree polynomials'', the [[Bell numbers]], {{math|''B''(''n'')}}, the [[Laguerre polynomials]], and the [[Stirling polynomial|Stirling convolution polynomials]].

===Tables of special generating functions===

An initial listing of special mathematical series is found [[List of mathematical series|here]]. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of ''Concrete Mathematics'' and in Section 2.5 of Wilf's ''Generatingfunctionology''. Other special generating functions of note include the entries in the next table, which is by no means complete.<ref>See also the ''1031 Generating Functions'' found in {{cite thesis |first=Simon |last=Plouffe |title=Approximations de séries génératrices et quelques conjectures |trans-title=Approximations of generating functions and a few conjectures |year=1992 |type=Masters |publisher=Université du Québec à Montréal |language=fr |arxiv=0911.4975}}</ref>

{{expand section|Lists of special and special sequence generating functions. The next table is a start|date=April 2017}}

:{| class="wikitable"
|-
! Formal power series !! Generating-function formula !! Notes
|-
| <math>\sum_{n = 0}^\infty \binom{m+n}{n} \left(H_{n+m}-H_m\right) z^n</math> || <math>\frac{1}{(1-z)^{m+1}} \ln \frac{1}{1-z}</math> || <math>H_n</math> is a first-order [[harmonic number]]
|-
| <math>\sum_{n = 0}^\infty B_n \frac{z^n}{n!}</math> || <math>\frac{z}{e^z-1}</math> || <math>B_n</math> is a [[Bernoulli number]]
|-
| <math>\sum_{n = 0}^\infty F_{mn} z^n</math> || <math>\frac{F_m z}{1-(F_{m-1}+F_{m+1})z+(-1)^m z^2}</math> || <math>F_n</math> is a [[Fibonacci number]] and <math>m \in \mathbb{Z}^{+}</math>
|-
| <math>\sum_{n = 0}^\infty \left\{\begin{matrix} n \\ m \end{matrix} \right\} z^n</math> || <math>(z^{-1})^{\overline{-m}} = \frac{z^m}{(1-z)(1-2z)\cdots(1-mz)}</math> || <math>x^{\overline{n}}</math> denotes the [[rising factorial]], or [[Pochhammer symbol]] and some integer <math>m \geq 0</math>
|-
| <math>\sum_{n = 0}^\infty \left[\begin{matrix} n \\ m \end{matrix} \right] z^n</math> || <math>z^{\overline{m}} = z(z+1) \cdots (z+m-1)</math>
|-
| <math>\sum_{n = 1}^\infty \frac{(-1)^{n-1}4^n (4^n-2) B_{2n} z^{2n}}{(2n) \cdot (2n)!}</math> || <math>\ln \frac{\tan(z)}{z}</math>
|-
| <math>\sum_{n = 0}^\infty \frac{(1/2)^{\overline{n}} z^{2n}}{(2n+1) \cdot n!}</math> || <math>z^{-1} \arcsin(z)</math>
|-
| <math>\sum_{n = 0}^\infty H_n^{(s)} z^n</math> || <math>\frac{\operatorname{Li}_s(z)}{1-z}</math> || <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]] function and <math>H_n^{(s)}</math> is a generalized [[harmonic number]] for <math>\Re(s) > 1</math>
|-
| <math>\sum_{n = 0}^\infty n^m z^n</math> || <math>\sum_{0 \leq j \leq m} \left\{\begin{matrix} m \\ j \end{matrix} \right\} \frac{j! \cdot z^j}{(1-z)^{j+1}}</math> || <math>\left\{\begin{matrix} n \\ m \end{matrix} \right\}</math> is a [[Stirling number of the second kind]] and where the individual terms in the expansion satisfy <math>\frac{z^i}{(1-z)^{i+1}} = \sum_{k=0}^{i} \binom{i}{k} \frac{(-1)^{k-i}}{(1-z)^{k+1}}</math>
|-
| <math>\sum_{k < n} \binom{n-k}{k} \frac{n}{n-k} z^k</math> || <math>\left(\frac{1+\sqrt{1+4z}}{2}\right)^n + \left(\frac{1-\sqrt{1+4z}}{2}\right)^n</math> ||
|-
| <math>\sum_{n_1, \ldots, n_m \geq 0} \min(n_1, \ldots, n_m) z_1^{n_1} \cdots z_m^{n_m}</math> || <math>\frac{z_1 \cdots z_m}{(1-z_1) \cdots (1-z_m) (1-z_1 \cdots z_m)}</math> || The two-variable case is given by <math>M(w, z) := \sum_{m,n \geq 0} \min(m, n) w^m z^n = \frac{wz}{(1-w)(1-z)(1-wz)}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{s}{n} z^n</math> || <math>(1+z)^s</math> || <math>s \in \mathbb{C}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{n}{k} z^n</math> || <math>\frac{z^k}{(1-z)^{k+1}}</math> || <math>k \in \mathbb{N}</math>
|-
|<math>\sum_{n = 1}^\infty \log{(n)} z^n</math>||<math>\left.-\frac{\partial}{\partial s}\operatorname{{Li}_s(z)}\right|_{s=0}</math>||
|}

== History ==
[[George Pólya]] writes in ''[[Mathematics and plausible reasoning]]'':
<blockquote>''The name "generating function" is due to [[Laplace]]. Yet, without giving it a name, [[Euler]] used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the [[Number theory|Theory of Numbers]].''</blockquote>

==See also==
* [[Moment-generating function]]
* [[Probability-generating function]]
* [[Generating function transformation]]
* [[Stanley's reciprocity theorem]]
* Applications to [[Partition (number theory)]]
* [[Combinatorial principles]]
* [[Cyclic sieving]]
* [[Z-transform]]
* [[Umbral calculus]]

==Notes==
{{noteFoot}}

==References==
{{reflist}}

===Citations===
*{{cite book |first=Martin |last=Aigner |title=A Course in Enumeration |url=https://books.google.com/books?id=pPEJcu93dzAC |date=2007 |publisher=Springer |isbn=978-3-540-39035-0 |series=Graduate Texts in Mathematics |volume=238 }}
* {{cite journal |title=On the foundations of combinatorial theory. VI. The idea of generating function |last1=Doubilet |first1=Peter |last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | last3=Stanley | first3=Richard | author3-link=Richard P. Stanley | journal=Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability |volume=2 |pages=267–318 |year=1972 | zbl=0267.05002 | url=http://projecteuclid.org/euclid.bsmsp/1200514223 }} Reprinted in {{cite book | last=Rota | first=Gian-Carlo | author-link=Gian-Carlo Rota | others=With the collaboration of P. Doubilet, C. Greene, D. Kahaner, [[Andrew Odlyzko|A. Odlyzko]] and [[Richard P. Stanley|R. Stanley]] | title=Finite Operator Calculus | chapter=3. The idea of generating function | pages=83–134 | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 | zbl=0328.05007 }}
* {{cite book | last1 = Flajolet | first1 = Philippe | author-link1 = Philippe Flajolet | last2 = Sedgewick | first2 = Robert | author-link2 = Robert Sedgewick (computer scientist) | title = Analytic Combinatorics | title-link= Analytic Combinatorics | year = 2009 | publisher = Cambridge University Press | isbn = 978-0-521-89806-5 | zbl=1165.05001 }}
* {{cite book | last1 = Goulden | first1 = Ian P. | last2 = Jackson | first2 = David M. | author-link2 = David M. Jackson | title = Combinatorial Enumeration | year = 2004 | publisher = [[Dover Publications]] | isbn = 978-0486435978 }}
* {{cite book |title=[[Concrete Mathematics|Concrete Mathematics. A foundation for computer science]] |edition=2nd |year=1994 |publisher=Addison-Wesley |isbn=0-201-55802-5 |chapter=Chapter 7: Generating Functions |pages=320–380| zbl=0836.00001 |first1 = Ronald L. |last1=Graham |first2 = Donald E. |last2=Knuth |first3=Oren |last3=Patashnik |author-link1=Ronald Graham |author-link2=Donald Knuth |author-link3=Oren Patashnik }}
*{{cite book |first=Sergei K. |last=Lando |title=Lectures on Generating Functions |url=https://books.google.com/books?id=A6_4AwAAQBAJ |date=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3481-7 }}
* {{cite book | last=Wilf | first=Herbert S. | author-link=Herbert Wilf | title=Generatingfunctionology | edition=2nd | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 | url=http://www.math.upenn.edu/%7Ewilf/DownldGF.html }}

==External links==
* [http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf "Introduction To Ordinary Generating Functions"] by Mike Zabrocki, York University, Mathematics and Statistics
* {{springer|title=Generating function|id=p/g043900}}
* [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/GeneratingFunctions/ "Generating Functions"] by [[Ed Pegg Jr.]], [[Wolfram Demonstrations Project]], 2007.

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Talk:Generating function

2023-11-25T21:26:57Z

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==References please==
Please give the references for the very nice formulas in the section on asymptotics of coefficients.
[[User:Asympt|Asympt]] ([[User talk:Asympt|talk]]) 18:57, 21 November 2021 (UTC)
:I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő, ''Problems and Theorems in Analysis, Vol 1.'' (1972), Exercise 174. And I see a citation to Wilf ''generatingfunctionology'' (1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 22:56, 21 November 2021 (UTC)

==Ancient comment==

The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
-[[User:Iopq|Iopq]] 19:59, 18 October 2005 (UTC)

== Definition ==

I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:

"A generation function is a transformation that converts a given sequence, ''S = {an}'', into a continous function, f(x), through a series expantion whose coeficients are the elements ''an'' of the sequence ''S''."

or something similar you find more appropiate.

:Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity. [[User:Charles Matthews|Charles Matthews]] 12:19, 16 November 2005 (UTC)

: I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the ''formal'' nature of the series. --[[User:Zero0000|Zero]] 22:52, 16 November 2005 (UTC)

::Absolutely. (No pun intended.) To call these things "continuous" is absurd. [[User:Michael Hardy|Michael Hardy]] 20:13, 17 November 2005 (UTC)

:::Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.[[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 14:45, 15 May 2019 (UTC)

:::: Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and [https://arxiv.org/abs/1106.5480 here] is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also [https://math.mit.edu/~rstan/ec/ec1.pdf EC1], notes on Chapter 1. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 15:20, 15 May 2019 (UTC)

::::: I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like <math>\Psi(x)</math> and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject. [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:54, 15 May 2019 (UTC)

I just noticed that the german version is not liked here it's called "Erzeugende Funktionen", url is here: http://de.wikipedia.org/wiki/Erzeugende_Funktion — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.254.173.61|80.254.173.61]] ([[User talk:80.254.173.61#top|talk]]) 16:44, 18 December 2005 (UTC)

== Examples please! ==

''In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. '''For example,'''...'' (a nice easy example or two, please!)

This article is fairly typical of current Wikipedia mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!

Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.

--[[User:84.9.78.198|84.9.78.198]] 14:14, 27 November 2006 (UTC)

:If you read on past the ''Definitions'' section you will find an ''Examples'' section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the [[Fibonacci number]]s is derived. If ''Examples'' came before ''Definitions'' the article would be more difficult to follow, as you would not know what the ''Examples'' were meant to be illustrating. [[User:Gandalf61|Gandalf61]] 14:41, 27 November 2006 (UTC)

== Uniqueness of F ==

I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-''X''). If ''F'' is any ring with a unit, not necessarily commutative or an integral domain, then the only power series <math>f(X) \in F[[X]]</math> such that <math>1=(1-X)f(X)</math> is <math>f(X)=1+X+X^2+\dots</math>. To see this, let <math> f(X) = f_0 + f_1 X + f_2 X^2 + \dots</math>. Then <math>(1-X)f(X) = f_0 + (f_1 - f_0) X + (f_2 - f_1) X^2 + ...</math>. Solving <math> 1 = (1-X)f(X) </math> means that <math> f_0 = 1, (f_1 - f_0) = 0, (f_2 - f_0) = 0, \dots</math> and therefore <math> 1 = f_0 = f_1 = f_2 = \dots </math>. The only multiplication in the ring ''F'' is used in this proof is multiplication by 1 and -1 in ''F''. Therefore neither general commutativity of the ring ''F'' nor ''F'' being an integral domain is required. Indeed multiplication in ''F'' need not even be associative. (So, the result holds if ''F'' is the octonions, for example.) It is only required that multiplication in ''F'' have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by ''F'' being a ring. Of course, multiplication by ''X'' in <math>F[[X]]</math> has been used, and commutativity of this operation , that is, <math>Xa = aX</math> has been used, as has the fact if <math>Xa=0</math> then <math>a=0</math>. In general, though if the ring of coefficient is not an integral domain or commutative, then neither is the resulting power series ring. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]] • [[Special:Contributions/DRLB|contribs]]) 15:26, 17 October 2008 (UTC) 

: That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --[[User:Charleyc|Charleyc]] ([[User talk:Charleyc|talk]]) 16:23, 18 October 2008 (UTC)

:: Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the article.) [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]]) 14:50, 20 October 2008 (UTC)

== Formulae ==

I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3... —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.187.35.160|85.187.35.160]] ([[User talk:85.187.35.160|talk]]) 13:47, 20 August 2009 (UTC) 

:Fixed - I have replaced <math>\sum_{n\in\mathbf{N}}</math> with <math>\sum_{n=0}^{\infty}</math>, which was clearly what was intended in each case. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:55, 20 August 2009 (UTC)

== "Generating series" terminology ==

The [http://en.wikipedia.org/w/index.php?title=Generating_function&oldid=364447959 current version] of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"? [[User:Quantling|Quantling]] ([[User talk:Quantling|talk]]) 16:00, 27 May 2010 (UTC)

:The "series" in "generating series" refers to [[formal power series]], where convergence is not much of an issue either (the term "generating formal power series" would be a bit heavy). Series are not necessarily about convergence, so I don't think this is much of a problem. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:35, 28 May 2010 (UTC)

== Is this a generating function? ==

<math>

\pi(\cot (\pi(c+z))-2\cot (2\pi(c-z))-2\cot (2\pi(c+z))+\cot (\pi(c-z)))

= -2 \left ( \sum_{k=0}^\infty z^{2k} \sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{2k+1}} - \frac{1}{(x-c-1/2)^{2k+1}} \right )

</math>

taking multiple derivatives with respect to z closed form sums can be obtained such as:

<math>

\pi^{3}(8(\cot(2c\pi)+\cot^{3}(2c\pi))-(\cot(c\pi)+\cot^{3}(c\pi))) =\sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{3}} - \frac{1}{(x-c-1/2)^{3}}

</math>

<math>

= -\frac{\pi^{3}\sin(c\pi)}{\cos^{3}(c\pi)}

</math>

<math>

\cot(\pi(c+z)) \approx \cot(c\pi)-z\pi(1+\cot^{2}(c\pi))+z^{2}\pi^{2}(\cot(c\pi)+\cot^{3}(c\pi))

</math>

http://iamned.com/math —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.161.40.148|67.161.40.148]] ([[User talk:67.161.40.148|talk]]) 11:02, 30 June 2010 (UTC) 

== confusing definition ==

The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing. [[User:RobLandau|RobLandau]] ([[User talk:RobLandau|talk]]) 07:40, 8 February 2018 (UTC)
:{{ping|RobLandau}} Generating functions are ''not'' sequences. The article does not say that; the article says they are used to ''describe'' sequences. I see no confusion here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:44, 8 February 2018 (UTC)
:: The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 13:05, 8 February 2018 (UTC)

Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which {{ping|Joel B. Lewis}} has twice restored, doesn’t make sense. The original and restored sentence ''The sum of this infinite series is the generating function'', as I said in my edit summary when I changed it and as RobLandau said above, is certain to give some people the impression that it means “The number that this series sums to is the generating function”, which is not right.

My replacement sentence, which I’m not wedded to, said ''The summation of this infinite series is the generating function''. Here ''[[summation]]'', as per its article, means ''the addition of a sequence of numbers'', which correctly refers to the entity rather than the result.

Joel restored the original with the edit summary ''The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)'' But many readers will not understand that here ''sum'' is intended to mean ''the whole infinite series''. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both ''sum'' and ''summation''. Thanks! [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 16:16, 29 May 2018 (UTC)

:I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using, ''This [[formal power series]] is the generating function.'' --[[User:Wcherowi|Bill Cherowitzo ]] ([[User talk:Wcherowi|talk]]) 17:04, 29 May 2018 (UTC)

:: Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is [https://en.wikipedia.org/w/index.php?title=Generating_function&diff=824616355&oldid=814318669 this one], which did not "restore" anything. The phrase "the summation of a series" makes no sense; if it did make sense, it would mean exactly the same as "the sum of the series". Indeed, the series ''is'' the sum; this sum is not a [real or complex] number because the individual summands are not [real or complex] numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:10, 29 May 2018 (UTC)
:::Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 19:31, 29 May 2018 (UTC)

:::: Your comment describes me has having "twice restored" something. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 29 May 2018 (UTC)

:::::Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 20:38, 29 May 2018 (UTC)
== Function* listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px|left]]
An editor has asked for a discussion to address the redirect [[Function*]]. Please participate in [[Wikipedia:Redirects for discussion/Log/2019 May 11#Function*|the redirect discussion]] if you wish to do so.  [[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:51, 11 May 2019 (UTC)

== Article is a mess ==

This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
* By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for Lambert, Bell, and formal Dirichlet series). The current version is trying to do ''way'' too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
* The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
* There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments".... [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:31, 15 May 2019 (UTC)

== must it be infinite? ==

Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial <math>(\frac{2}{5}x^1 + \frac{1}{5}x^2 + \frac{1}{5}x^3 + \frac{1}{5}x^4)^{64} </math>. Is that not a generating function because it's not infinite? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 14:56, 16 October 2019 (UTC)
: Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:17, 16 October 2019 (UTC)
:: The wiki-linking in the lede is also rather [[WP:SUBMARINE|submarine]]. It links to [[formal power series]] with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to [[formal power series]] ''again'' with the text "formal series". Next we get {{tq|Generating functions were first introduced by Abraham de Moivre in 1730}} — fine — {{tq| in order to solve the general linear recurrence problem.}} Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence problem just the challenge of understanding linear recurrences in general? [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 05:21, 17 October 2019 (UTC)

== Formula for generating function for a linear recursive sequrnce. ==

The following formula is really easy to use. Shall it be included in this article?

Let <math>s_n</math> be a linear recursive sequence of order k with initial conditions
<math> \{s_0, s_1, \ldots, s_{k-1}\}</math> and recursive relation
<math>s_n = \sum_{i=1}^k a_i s_{n-i}.</math>

Then the generating function for $s_n$ is given by the formula

<math>(\sum_{i=0}^{k-1} ( \sum_{j=0}^{i} (-a_j)* s_{i-j}) * x^{i-k})/f(x^{-1})</math> — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Kaiwang45|Kaiwang45]] ([[User talk:Kaiwang45#top|talk]] • [[Special:Contributions/Kaiwang45|contribs]]) 15:49, 27 July 2020 (UTC) 

== Blackboard bold formatting ==

{{reply|Quantling}} Greetings! Regarding [https://en.wikipedia.org/w/index.php?title=Generating_function&oldid=prev&diff=1146738276&diffmode=source this revert]...the use of {{tag|math}} is required by [[MOS:BBB]]. If we want the nearby markup to be consistent, that's fine; we would just need to convert it to also use {{tag|math}}. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 16:21, 27 March 2023 (UTC)
:{{reply to|Beland}} Good point. To be more consistent with [[MOS:STYLERET]], other possibilities are to use
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'', ''b'' ∈ '''N'''}}, {{math|''a'' ≥ 2}}, and {{math|0 ≤ ''b'' < ''a''}}.
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}} and {{math|0 ≤ ''b'' < ''a''}} are integers.
:What do you think? —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:39, 27 March 2023 (UTC)
::{{reply|Quantling}} "{{math|''a''}} and {{math|''b''}} are integers" is certainly a lot less jargony than using the blackboard bold notation. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 17:45, 27 March 2023 (UTC)
:::I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:55, 27 March 2023 (UTC)
::::Done; thanks for your help ironing this out! -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 22:09, 27 March 2023 (UTC)

== Remove Sections ==

It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?

For starters, we should probably remove P-holonomic functions and J-fractions and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.

Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! (Ex: The 'Article is a mess' post above seems rather insightful, and I'll try to propose concrete edits which might circumvent the proposed issues.)

Also, how about this one: We just list a couple applications of generating functions (I honestly think snake oil or something is a good enough thing to convince people that they're 'useful', and then maintain a 'main article' on applications). I wish to scrap off the entire J-fraction part, write something about them in a main link, write about transforming between ordinary and exponential generating functions and then remove the whole transforming part.

[[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 03:27, 25 November 2023 (UTC)

Talk:Generating function

2023-11-25T03:36:53Z

Yeetcode: /* Remove Sections */

{{Vital article|class=C|level=5|topic=Mathematics}}
{{maths rating
|field = discrete
|importance = high
|class = C
|historical =
}}
{{Broken anchors|links=
* <nowiki>[[Geometric_series#Closed-form_formula|geometric series]]</nowiki> The anchor (#Closed-form_formula) has been [[Special:Diff/1129004581|deleted by other users]] before. 
}}

==References please==
Please give the references for the very nice formulas in the section on asymptotics of coefficients.
[[User:Asympt|Asympt]] ([[User talk:Asympt|talk]]) 18:57, 21 November 2021 (UTC)
:I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő, ''Problems and Theorems in Analysis, Vol 1.'' (1972), Exercise 174. And I see a citation to Wilf ''generatingfunctionology'' (1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 22:56, 21 November 2021 (UTC)

==Ancient comment==

The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
-[[User:Iopq|Iopq]] 19:59, 18 October 2005 (UTC)

== Definition ==

I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:

"A generation function is a transformation that converts a given sequence, ''S = {an}'', into a continous function, f(x), through a series expantion whose coeficients are the elements ''an'' of the sequence ''S''."

or something similar you find more appropiate.

:Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity. [[User:Charles Matthews|Charles Matthews]] 12:19, 16 November 2005 (UTC)

: I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the ''formal'' nature of the series. --[[User:Zero0000|Zero]] 22:52, 16 November 2005 (UTC)

::Absolutely. (No pun intended.) To call these things "continuous" is absurd. [[User:Michael Hardy|Michael Hardy]] 20:13, 17 November 2005 (UTC)

:::Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.[[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 14:45, 15 May 2019 (UTC)

:::: Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and [https://arxiv.org/abs/1106.5480 here] is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also [https://math.mit.edu/~rstan/ec/ec1.pdf EC1], notes on Chapter 1. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 15:20, 15 May 2019 (UTC)

::::: I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like <math>\Psi(x)</math> and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject. [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:54, 15 May 2019 (UTC)

I just noticed that the german version is not liked here it's called "Erzeugende Funktionen", url is here: http://de.wikipedia.org/wiki/Erzeugende_Funktion — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.254.173.61|80.254.173.61]] ([[User talk:80.254.173.61#top|talk]]) 16:44, 18 December 2005 (UTC)

== Examples please! ==

''In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. '''For example,'''...'' (a nice easy example or two, please!)

This article is fairly typical of current Wikipedia mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!

Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.

--[[User:84.9.78.198|84.9.78.198]] 14:14, 27 November 2006 (UTC)

:If you read on past the ''Definitions'' section you will find an ''Examples'' section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the [[Fibonacci number]]s is derived. If ''Examples'' came before ''Definitions'' the article would be more difficult to follow, as you would not know what the ''Examples'' were meant to be illustrating. [[User:Gandalf61|Gandalf61]] 14:41, 27 November 2006 (UTC)

== Uniqueness of F ==

I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-''X''). If ''F'' is any ring with a unit, not necessarily commutative or an integral domain, then the only power series <math>f(X) \in F[[X]]</math> such that <math>1=(1-X)f(X)</math> is <math>f(X)=1+X+X^2+\dots</math>. To see this, let <math> f(X) = f_0 + f_1 X + f_2 X^2 + \dots</math>. Then <math>(1-X)f(X) = f_0 + (f_1 - f_0) X + (f_2 - f_1) X^2 + ...</math>. Solving <math> 1 = (1-X)f(X) </math> means that <math> f_0 = 1, (f_1 - f_0) = 0, (f_2 - f_0) = 0, \dots</math> and therefore <math> 1 = f_0 = f_1 = f_2 = \dots </math>. The only multiplication in the ring ''F'' is used in this proof is multiplication by 1 and -1 in ''F''. Therefore neither general commutativity of the ring ''F'' nor ''F'' being an integral domain is required. Indeed multiplication in ''F'' need not even be associative. (So, the result holds if ''F'' is the octonions, for example.) It is only required that multiplication in ''F'' have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by ''F'' being a ring. Of course, multiplication by ''X'' in <math>F[[X]]</math> has been used, and commutativity of this operation , that is, <math>Xa = aX</math> has been used, as has the fact if <math>Xa=0</math> then <math>a=0</math>. In general, though if the ring of coefficient is not an integral domain or commutative, then neither is the resulting power series ring. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]] • [[Special:Contributions/DRLB|contribs]]) 15:26, 17 October 2008 (UTC) 

: That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --[[User:Charleyc|Charleyc]] ([[User talk:Charleyc|talk]]) 16:23, 18 October 2008 (UTC)

:: Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the article.) [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]]) 14:50, 20 October 2008 (UTC)

== Formulae ==

I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3... —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.187.35.160|85.187.35.160]] ([[User talk:85.187.35.160|talk]]) 13:47, 20 August 2009 (UTC) 

:Fixed - I have replaced <math>\sum_{n\in\mathbf{N}}</math> with <math>\sum_{n=0}^{\infty}</math>, which was clearly what was intended in each case. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:55, 20 August 2009 (UTC)

== "Generating series" terminology ==

The [http://en.wikipedia.org/w/index.php?title=Generating_function&oldid=364447959 current version] of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"? [[User:Quantling|Quantling]] ([[User talk:Quantling|talk]]) 16:00, 27 May 2010 (UTC)

:The "series" in "generating series" refers to [[formal power series]], where convergence is not much of an issue either (the term "generating formal power series" would be a bit heavy). Series are not necessarily about convergence, so I don't think this is much of a problem. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:35, 28 May 2010 (UTC)

== Is this a generating function? ==

<math>

\pi(\cot (\pi(c+z))-2\cot (2\pi(c-z))-2\cot (2\pi(c+z))+\cot (\pi(c-z)))

= -2 \left ( \sum_{k=0}^\infty z^{2k} \sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{2k+1}} - \frac{1}{(x-c-1/2)^{2k+1}} \right )

</math>

taking multiple derivatives with respect to z closed form sums can be obtained such as:

<math>

\pi^{3}(8(\cot(2c\pi)+\cot^{3}(2c\pi))-(\cot(c\pi)+\cot^{3}(c\pi))) =\sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{3}} - \frac{1}{(x-c-1/2)^{3}}

</math>

<math>

= -\frac{\pi^{3}\sin(c\pi)}{\cos^{3}(c\pi)}

</math>

<math>

\cot(\pi(c+z)) \approx \cot(c\pi)-z\pi(1+\cot^{2}(c\pi))+z^{2}\pi^{2}(\cot(c\pi)+\cot^{3}(c\pi))

</math>

http://iamned.com/math —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.161.40.148|67.161.40.148]] ([[User talk:67.161.40.148|talk]]) 11:02, 30 June 2010 (UTC) 

== confusing definition ==

The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing. [[User:RobLandau|RobLandau]] ([[User talk:RobLandau|talk]]) 07:40, 8 February 2018 (UTC)
:{{ping|RobLandau}} Generating functions are ''not'' sequences. The article does not say that; the article says they are used to ''describe'' sequences. I see no confusion here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:44, 8 February 2018 (UTC)
:: The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 13:05, 8 February 2018 (UTC)

Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which {{ping|Joel B. Lewis}} has twice restored, doesn’t make sense. The original and restored sentence ''The sum of this infinite series is the generating function'', as I said in my edit summary when I changed it and as RobLandau said above, is certain to give some people the impression that it means “The number that this series sums to is the generating function”, which is not right.

My replacement sentence, which I’m not wedded to, said ''The summation of this infinite series is the generating function''. Here ''[[summation]]'', as per its article, means ''the addition of a sequence of numbers'', which correctly refers to the entity rather than the result.

Joel restored the original with the edit summary ''The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)'' But many readers will not understand that here ''sum'' is intended to mean ''the whole infinite series''. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both ''sum'' and ''summation''. Thanks! [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 16:16, 29 May 2018 (UTC)

:I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using, ''This [[formal power series]] is the generating function.'' --[[User:Wcherowi|Bill Cherowitzo ]] ([[User talk:Wcherowi|talk]]) 17:04, 29 May 2018 (UTC)

:: Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is [https://en.wikipedia.org/w/index.php?title=Generating_function&diff=824616355&oldid=814318669 this one], which did not "restore" anything. The phrase "the summation of a series" makes no sense; if it did make sense, it would mean exactly the same as "the sum of the series". Indeed, the series ''is'' the sum; this sum is not a [real or complex] number because the individual summands are not [real or complex] numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:10, 29 May 2018 (UTC)
:::Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 19:31, 29 May 2018 (UTC)

:::: Your comment describes me has having "twice restored" something. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 29 May 2018 (UTC)

:::::Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 20:38, 29 May 2018 (UTC)
== Function* listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px|left]]
An editor has asked for a discussion to address the redirect [[Function*]]. Please participate in [[Wikipedia:Redirects for discussion/Log/2019 May 11#Function*|the redirect discussion]] if you wish to do so.  [[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:51, 11 May 2019 (UTC)

== Article is a mess ==

This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
* By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for Lambert, Bell, and formal Dirichlet series). The current version is trying to do ''way'' too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
* The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
* There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments".... [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:31, 15 May 2019 (UTC)

== must it be infinite? ==

Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial <math>(\frac{2}{5}x^1 + \frac{1}{5}x^2 + \frac{1}{5}x^3 + \frac{1}{5}x^4)^{64} </math>. Is that not a generating function because it's not infinite? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 14:56, 16 October 2019 (UTC)
: Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:17, 16 October 2019 (UTC)
:: The wiki-linking in the lede is also rather [[WP:SUBMARINE|submarine]]. It links to [[formal power series]] with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to [[formal power series]] ''again'' with the text "formal series". Next we get {{tq|Generating functions were first introduced by Abraham de Moivre in 1730}} — fine — {{tq| in order to solve the general linear recurrence problem.}} Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence problem just the challenge of understanding linear recurrences in general? [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 05:21, 17 October 2019 (UTC)

== Formula for generating function for a linear recursive sequrnce. ==

The following formula is really easy to use. Shall it be included in this article?

Let <math>s_n</math> be a linear recursive sequence of order k with initial conditions
<math> \{s_0, s_1, \ldots, s_{k-1}\}</math> and recursive relation
<math>s_n = \sum_{i=1}^k a_i s_{n-i}.</math>

Then the generating function for $s_n$ is given by the formula

<math>(\sum_{i=0}^{k-1} ( \sum_{j=0}^{i} (-a_j)* s_{i-j}) * x^{i-k})/f(x^{-1})</math> — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Kaiwang45|Kaiwang45]] ([[User talk:Kaiwang45#top|talk]] • [[Special:Contributions/Kaiwang45|contribs]]) 15:49, 27 July 2020 (UTC) 

== Blackboard bold formatting ==

{{reply|Quantling}} Greetings! Regarding [https://en.wikipedia.org/w/index.php?title=Generating_function&oldid=prev&diff=1146738276&diffmode=source this revert]...the use of {{tag|math}} is required by [[MOS:BBB]]. If we want the nearby markup to be consistent, that's fine; we would just need to convert it to also use {{tag|math}}. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 16:21, 27 March 2023 (UTC)
:{{reply to|Beland}} Good point. To be more consistent with [[MOS:STYLERET]], other possibilities are to use
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'', ''b'' ∈ '''N'''}}, {{math|''a'' ≥ 2}}, and {{math|0 ≤ ''b'' < ''a''}}.
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}} and {{math|0 ≤ ''b'' < ''a''}} are integers.
:What do you think? —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:39, 27 March 2023 (UTC)
::{{reply|Quantling}} "{{math|''a''}} and {{math|''b''}} are integers" is certainly a lot less jargony than using the blackboard bold notation. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 17:45, 27 March 2023 (UTC)
:::I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:55, 27 March 2023 (UTC)
::::Done; thanks for your help ironing this out! -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 22:09, 27 March 2023 (UTC)

== Remove Sections ==

It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?

For starters, we should probably remove P-holonomic functions and J-representations and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.

Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! (Ex: The 'Article is a mess' post above seems rather insightful, and I'll try to propose concrete edits which might circumvent the proposed issues.)

Also, how about this one: We just list a couple applications of generating functions (I honestly think snake oil or something is a good enough thing to convince people that they're 'useful', and then maintain a 'main article' on applications).

[[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 03:27, 25 November 2023 (UTC)

Talk:Generating function

2023-11-25T03:30:02Z

Yeetcode: /* Remove Sections */

{{Vital article|class=C|level=5|topic=Mathematics}}
{{maths rating
|field = discrete
|importance = high
|class = C
|historical =
}}
{{Broken anchors|links=
* <nowiki>[[Geometric_series#Closed-form_formula|geometric series]]</nowiki> The anchor (#Closed-form_formula) has been [[Special:Diff/1129004581|deleted by other users]] before. 
}}

==References please==
Please give the references for the very nice formulas in the section on asymptotics of coefficients.
[[User:Asympt|Asympt]] ([[User talk:Asympt|talk]]) 18:57, 21 November 2021 (UTC)
:I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő, ''Problems and Theorems in Analysis, Vol 1.'' (1972), Exercise 174. And I see a citation to Wilf ''generatingfunctionology'' (1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 22:56, 21 November 2021 (UTC)

==Ancient comment==

The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
-[[User:Iopq|Iopq]] 19:59, 18 October 2005 (UTC)

== Definition ==

I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:

"A generation function is a transformation that converts a given sequence, ''S = {an}'', into a continous function, f(x), through a series expantion whose coeficients are the elements ''an'' of the sequence ''S''."

or something similar you find more appropiate.

:Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity. [[User:Charles Matthews|Charles Matthews]] 12:19, 16 November 2005 (UTC)

: I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the ''formal'' nature of the series. --[[User:Zero0000|Zero]] 22:52, 16 November 2005 (UTC)

::Absolutely. (No pun intended.) To call these things "continuous" is absurd. [[User:Michael Hardy|Michael Hardy]] 20:13, 17 November 2005 (UTC)

:::Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.[[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 14:45, 15 May 2019 (UTC)

:::: Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and [https://arxiv.org/abs/1106.5480 here] is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also [https://math.mit.edu/~rstan/ec/ec1.pdf EC1], notes on Chapter 1. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 15:20, 15 May 2019 (UTC)

::::: I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like <math>\Psi(x)</math> and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject. [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:54, 15 May 2019 (UTC)

I just noticed that the german version is not liked here it's called "Erzeugende Funktionen", url is here: http://de.wikipedia.org/wiki/Erzeugende_Funktion — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.254.173.61|80.254.173.61]] ([[User talk:80.254.173.61#top|talk]]) 16:44, 18 December 2005 (UTC)

== Examples please! ==

''In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. '''For example,'''...'' (a nice easy example or two, please!)

This article is fairly typical of current Wikipedia mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!

Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.

--[[User:84.9.78.198|84.9.78.198]] 14:14, 27 November 2006 (UTC)

:If you read on past the ''Definitions'' section you will find an ''Examples'' section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the [[Fibonacci number]]s is derived. If ''Examples'' came before ''Definitions'' the article would be more difficult to follow, as you would not know what the ''Examples'' were meant to be illustrating. [[User:Gandalf61|Gandalf61]] 14:41, 27 November 2006 (UTC)

== Uniqueness of F ==

I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-''X''). If ''F'' is any ring with a unit, not necessarily commutative or an integral domain, then the only power series <math>f(X) \in F[[X]]</math> such that <math>1=(1-X)f(X)</math> is <math>f(X)=1+X+X^2+\dots</math>. To see this, let <math> f(X) = f_0 + f_1 X + f_2 X^2 + \dots</math>. Then <math>(1-X)f(X) = f_0 + (f_1 - f_0) X + (f_2 - f_1) X^2 + ...</math>. Solving <math> 1 = (1-X)f(X) </math> means that <math> f_0 = 1, (f_1 - f_0) = 0, (f_2 - f_0) = 0, \dots</math> and therefore <math> 1 = f_0 = f_1 = f_2 = \dots </math>. The only multiplication in the ring ''F'' is used in this proof is multiplication by 1 and -1 in ''F''. Therefore neither general commutativity of the ring ''F'' nor ''F'' being an integral domain is required. Indeed multiplication in ''F'' need not even be associative. (So, the result holds if ''F'' is the octonions, for example.) It is only required that multiplication in ''F'' have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by ''F'' being a ring. Of course, multiplication by ''X'' in <math>F[[X]]</math> has been used, and commutativity of this operation , that is, <math>Xa = aX</math> has been used, as has the fact if <math>Xa=0</math> then <math>a=0</math>. In general, though if the ring of coefficient is not an integral domain or commutative, then neither is the resulting power series ring. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]] • [[Special:Contributions/DRLB|contribs]]) 15:26, 17 October 2008 (UTC) 

: That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --[[User:Charleyc|Charleyc]] ([[User talk:Charleyc|talk]]) 16:23, 18 October 2008 (UTC)

:: Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the article.) [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]]) 14:50, 20 October 2008 (UTC)

== Formulae ==

I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3... —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.187.35.160|85.187.35.160]] ([[User talk:85.187.35.160|talk]]) 13:47, 20 August 2009 (UTC) 

:Fixed - I have replaced <math>\sum_{n\in\mathbf{N}}</math> with <math>\sum_{n=0}^{\infty}</math>, which was clearly what was intended in each case. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:55, 20 August 2009 (UTC)

== "Generating series" terminology ==

The [http://en.wikipedia.org/w/index.php?title=Generating_function&oldid=364447959 current version] of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"? [[User:Quantling|Quantling]] ([[User talk:Quantling|talk]]) 16:00, 27 May 2010 (UTC)

:The "series" in "generating series" refers to [[formal power series]], where convergence is not much of an issue either (the term "generating formal power series" would be a bit heavy). Series are not necessarily about convergence, so I don't think this is much of a problem. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:35, 28 May 2010 (UTC)

== Is this a generating function? ==

<math>

\pi(\cot (\pi(c+z))-2\cot (2\pi(c-z))-2\cot (2\pi(c+z))+\cot (\pi(c-z)))

= -2 \left ( \sum_{k=0}^\infty z^{2k} \sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{2k+1}} - \frac{1}{(x-c-1/2)^{2k+1}} \right )

</math>

taking multiple derivatives with respect to z closed form sums can be obtained such as:

<math>

\pi^{3}(8(\cot(2c\pi)+\cot^{3}(2c\pi))-(\cot(c\pi)+\cot^{3}(c\pi))) =\sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{3}} - \frac{1}{(x-c-1/2)^{3}}

</math>

<math>

= -\frac{\pi^{3}\sin(c\pi)}{\cos^{3}(c\pi)}

</math>

<math>

\cot(\pi(c+z)) \approx \cot(c\pi)-z\pi(1+\cot^{2}(c\pi))+z^{2}\pi^{2}(\cot(c\pi)+\cot^{3}(c\pi))

</math>

http://iamned.com/math —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.161.40.148|67.161.40.148]] ([[User talk:67.161.40.148|talk]]) 11:02, 30 June 2010 (UTC) 

== confusing definition ==

The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing. [[User:RobLandau|RobLandau]] ([[User talk:RobLandau|talk]]) 07:40, 8 February 2018 (UTC)
:{{ping|RobLandau}} Generating functions are ''not'' sequences. The article does not say that; the article says they are used to ''describe'' sequences. I see no confusion here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:44, 8 February 2018 (UTC)
:: The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 13:05, 8 February 2018 (UTC)

Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which {{ping|Joel B. Lewis}} has twice restored, doesn’t make sense. The original and restored sentence ''The sum of this infinite series is the generating function'', as I said in my edit summary when I changed it and as RobLandau said above, is certain to give some people the impression that it means “The number that this series sums to is the generating function”, which is not right.

My replacement sentence, which I’m not wedded to, said ''The summation of this infinite series is the generating function''. Here ''[[summation]]'', as per its article, means ''the addition of a sequence of numbers'', which correctly refers to the entity rather than the result.

Joel restored the original with the edit summary ''The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)'' But many readers will not understand that here ''sum'' is intended to mean ''the whole infinite series''. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both ''sum'' and ''summation''. Thanks! [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 16:16, 29 May 2018 (UTC)

:I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using, ''This [[formal power series]] is the generating function.'' --[[User:Wcherowi|Bill Cherowitzo ]] ([[User talk:Wcherowi|talk]]) 17:04, 29 May 2018 (UTC)

:: Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is [https://en.wikipedia.org/w/index.php?title=Generating_function&diff=824616355&oldid=814318669 this one], which did not "restore" anything. The phrase "the summation of a series" makes no sense; if it did make sense, it would mean exactly the same as "the sum of the series". Indeed, the series ''is'' the sum; this sum is not a [real or complex] number because the individual summands are not [real or complex] numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:10, 29 May 2018 (UTC)
:::Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 19:31, 29 May 2018 (UTC)

:::: Your comment describes me has having "twice restored" something. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 29 May 2018 (UTC)

:::::Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 20:38, 29 May 2018 (UTC)
== Function* listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px|left]]
An editor has asked for a discussion to address the redirect [[Function*]]. Please participate in [[Wikipedia:Redirects for discussion/Log/2019 May 11#Function*|the redirect discussion]] if you wish to do so.  [[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:51, 11 May 2019 (UTC)

== Article is a mess ==

This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
* By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for Lambert, Bell, and formal Dirichlet series). The current version is trying to do ''way'' too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
* The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
* There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments".... [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:31, 15 May 2019 (UTC)

== must it be infinite? ==

Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial <math>(\frac{2}{5}x^1 + \frac{1}{5}x^2 + \frac{1}{5}x^3 + \frac{1}{5}x^4)^{64} </math>. Is that not a generating function because it's not infinite? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 14:56, 16 October 2019 (UTC)
: Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:17, 16 October 2019 (UTC)
:: The wiki-linking in the lede is also rather [[WP:SUBMARINE|submarine]]. It links to [[formal power series]] with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to [[formal power series]] ''again'' with the text "formal series". Next we get {{tq|Generating functions were first introduced by Abraham de Moivre in 1730}} — fine — {{tq| in order to solve the general linear recurrence problem.}} Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence problem just the challenge of understanding linear recurrences in general? [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 05:21, 17 October 2019 (UTC)

== Formula for generating function for a linear recursive sequrnce. ==

The following formula is really easy to use. Shall it be included in this article?

Let <math>s_n</math> be a linear recursive sequence of order k with initial conditions
<math> \{s_0, s_1, \ldots, s_{k-1}\}</math> and recursive relation
<math>s_n = \sum_{i=1}^k a_i s_{n-i}.</math>

Then the generating function for $s_n$ is given by the formula

<math>(\sum_{i=0}^{k-1} ( \sum_{j=0}^{i} (-a_j)* s_{i-j}) * x^{i-k})/f(x^{-1})</math> — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Kaiwang45|Kaiwang45]] ([[User talk:Kaiwang45#top|talk]] • [[Special:Contributions/Kaiwang45|contribs]]) 15:49, 27 July 2020 (UTC) 

== Blackboard bold formatting ==

{{reply|Quantling}} Greetings! Regarding [https://en.wikipedia.org/w/index.php?title=Generating_function&oldid=prev&diff=1146738276&diffmode=source this revert]...the use of {{tag|math}} is required by [[MOS:BBB]]. If we want the nearby markup to be consistent, that's fine; we would just need to convert it to also use {{tag|math}}. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 16:21, 27 March 2023 (UTC)
:{{reply to|Beland}} Good point. To be more consistent with [[MOS:STYLERET]], other possibilities are to use
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'', ''b'' ∈ '''N'''}}, {{math|''a'' ≥ 2}}, and {{math|0 ≤ ''b'' < ''a''}}.
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}} and {{math|0 ≤ ''b'' < ''a''}} are integers.
:What do you think? —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:39, 27 March 2023 (UTC)
::{{reply|Quantling}} "{{math|''a''}} and {{math|''b''}} are integers" is certainly a lot less jargony than using the blackboard bold notation. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 17:45, 27 March 2023 (UTC)
:::I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:55, 27 March 2023 (UTC)
::::Done; thanks for your help ironing this out! -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 22:09, 27 March 2023 (UTC)

== Remove Sections ==

It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?

For starters, we should probably remove P-holonomic functions and J-representations and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.

Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! (Ex: The 'Article is a mess' post above seems rather insightful, and I'll try to propose concrete edits which might circumvent the proposed issues.)[[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 03:27, 25 November 2023 (UTC)

Talk:Generating function

2023-11-25T03:27:52Z

Yeetcode: /* Remove Sections */ new section

{{Vital article|class=C|level=5|topic=Mathematics}}
{{maths rating
|field = discrete
|importance = high
|class = C
|historical =
}}
{{Broken anchors|links=
* <nowiki>[[Geometric_series#Closed-form_formula|geometric series]]</nowiki> The anchor (#Closed-form_formula) has been [[Special:Diff/1129004581|deleted by other users]] before. 
}}

==References please==
Please give the references for the very nice formulas in the section on asymptotics of coefficients.
[[User:Asympt|Asympt]] ([[User talk:Asympt|talk]]) 18:57, 21 November 2021 (UTC)
:I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő, ''Problems and Theorems in Analysis, Vol 1.'' (1972), Exercise 174. And I see a citation to Wilf ''generatingfunctionology'' (1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 22:56, 21 November 2021 (UTC)

==Ancient comment==

The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
-[[User:Iopq|Iopq]] 19:59, 18 October 2005 (UTC)

== Definition ==

I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:

"A generation function is a transformation that converts a given sequence, ''S = {an}'', into a continous function, f(x), through a series expantion whose coeficients are the elements ''an'' of the sequence ''S''."

or something similar you find more appropiate.

:Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity. [[User:Charles Matthews|Charles Matthews]] 12:19, 16 November 2005 (UTC)

: I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the ''formal'' nature of the series. --[[User:Zero0000|Zero]] 22:52, 16 November 2005 (UTC)

::Absolutely. (No pun intended.) To call these things "continuous" is absurd. [[User:Michael Hardy|Michael Hardy]] 20:13, 17 November 2005 (UTC)

:::Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.[[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 14:45, 15 May 2019 (UTC)

:::: Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and [https://arxiv.org/abs/1106.5480 here] is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also [https://math.mit.edu/~rstan/ec/ec1.pdf EC1], notes on Chapter 1. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 15:20, 15 May 2019 (UTC)

::::: I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like <math>\Psi(x)</math> and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject. [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:54, 15 May 2019 (UTC)

I just noticed that the german version is not liked here it's called "Erzeugende Funktionen", url is here: http://de.wikipedia.org/wiki/Erzeugende_Funktion — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.254.173.61|80.254.173.61]] ([[User talk:80.254.173.61#top|talk]]) 16:44, 18 December 2005 (UTC)

== Examples please! ==

''In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. '''For example,'''...'' (a nice easy example or two, please!)

This article is fairly typical of current Wikipedia mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!

Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.

--[[User:84.9.78.198|84.9.78.198]] 14:14, 27 November 2006 (UTC)

:If you read on past the ''Definitions'' section you will find an ''Examples'' section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the [[Fibonacci number]]s is derived. If ''Examples'' came before ''Definitions'' the article would be more difficult to follow, as you would not know what the ''Examples'' were meant to be illustrating. [[User:Gandalf61|Gandalf61]] 14:41, 27 November 2006 (UTC)

== Uniqueness of F ==

I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-''X''). If ''F'' is any ring with a unit, not necessarily commutative or an integral domain, then the only power series <math>f(X) \in F[[X]]</math> such that <math>1=(1-X)f(X)</math> is <math>f(X)=1+X+X^2+\dots</math>. To see this, let <math> f(X) = f_0 + f_1 X + f_2 X^2 + \dots</math>. Then <math>(1-X)f(X) = f_0 + (f_1 - f_0) X + (f_2 - f_1) X^2 + ...</math>. Solving <math> 1 = (1-X)f(X) </math> means that <math> f_0 = 1, (f_1 - f_0) = 0, (f_2 - f_0) = 0, \dots</math> and therefore <math> 1 = f_0 = f_1 = f_2 = \dots </math>. The only multiplication in the ring ''F'' is used in this proof is multiplication by 1 and -1 in ''F''. Therefore neither general commutativity of the ring ''F'' nor ''F'' being an integral domain is required. Indeed multiplication in ''F'' need not even be associative. (So, the result holds if ''F'' is the octonions, for example.) It is only required that multiplication in ''F'' have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by ''F'' being a ring. Of course, multiplication by ''X'' in <math>F[[X]]</math> has been used, and commutativity of this operation , that is, <math>Xa = aX</math> has been used, as has the fact if <math>Xa=0</math> then <math>a=0</math>. In general, though if the ring of coefficient is not an integral domain or commutative, then neither is the resulting power series ring. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]] • [[Special:Contributions/DRLB|contribs]]) 15:26, 17 October 2008 (UTC) 

: That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --[[User:Charleyc|Charleyc]] ([[User talk:Charleyc|talk]]) 16:23, 18 October 2008 (UTC)

:: Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the article.) [[User:DRLB|DRLB]] ([[User talk:DRLB|talk]]) 14:50, 20 October 2008 (UTC)

== Formulae ==

I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3... —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.187.35.160|85.187.35.160]] ([[User talk:85.187.35.160|talk]]) 13:47, 20 August 2009 (UTC) 

:Fixed - I have replaced <math>\sum_{n\in\mathbf{N}}</math> with <math>\sum_{n=0}^{\infty}</math>, which was clearly what was intended in each case. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:55, 20 August 2009 (UTC)

== "Generating series" terminology ==

The [http://en.wikipedia.org/w/index.php?title=Generating_function&oldid=364447959 current version] of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"? [[User:Quantling|Quantling]] ([[User talk:Quantling|talk]]) 16:00, 27 May 2010 (UTC)

:The "series" in "generating series" refers to [[formal power series]], where convergence is not much of an issue either (the term "generating formal power series" would be a bit heavy). Series are not necessarily about convergence, so I don't think this is much of a problem. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:35, 28 May 2010 (UTC)

== Is this a generating function? ==

<math>

\pi(\cot (\pi(c+z))-2\cot (2\pi(c-z))-2\cot (2\pi(c+z))+\cot (\pi(c-z)))

= -2 \left ( \sum_{k=0}^\infty z^{2k} \sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{2k+1}} - \frac{1}{(x-c-1/2)^{2k+1}} \right )

</math>

taking multiple derivatives with respect to z closed form sums can be obtained such as:

<math>

\pi^{3}(8(\cot(2c\pi)+\cot^{3}(2c\pi))-(\cot(c\pi)+\cot^{3}(c\pi))) =\sum_{x=1}^\infty \frac{1}{(x+c-1/2)^{3}} - \frac{1}{(x-c-1/2)^{3}}

</math>

<math>

= -\frac{\pi^{3}\sin(c\pi)}{\cos^{3}(c\pi)}

</math>

<math>

\cot(\pi(c+z)) \approx \cot(c\pi)-z\pi(1+\cot^{2}(c\pi))+z^{2}\pi^{2}(\cot(c\pi)+\cot^{3}(c\pi))

</math>

http://iamned.com/math —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.161.40.148|67.161.40.148]] ([[User talk:67.161.40.148|talk]]) 11:02, 30 June 2010 (UTC) 

== confusing definition ==

The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing. [[User:RobLandau|RobLandau]] ([[User talk:RobLandau|talk]]) 07:40, 8 February 2018 (UTC)
:{{ping|RobLandau}} Generating functions are ''not'' sequences. The article does not say that; the article says they are used to ''describe'' sequences. I see no confusion here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:44, 8 February 2018 (UTC)
:: The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 13:05, 8 February 2018 (UTC)

Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which {{ping|Joel B. Lewis}} has twice restored, doesn’t make sense. The original and restored sentence ''The sum of this infinite series is the generating function'', as I said in my edit summary when I changed it and as RobLandau said above, is certain to give some people the impression that it means “The number that this series sums to is the generating function”, which is not right.

My replacement sentence, which I’m not wedded to, said ''The summation of this infinite series is the generating function''. Here ''[[summation]]'', as per its article, means ''the addition of a sequence of numbers'', which correctly refers to the entity rather than the result.

Joel restored the original with the edit summary ''The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)'' But many readers will not understand that here ''sum'' is intended to mean ''the whole infinite series''. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both ''sum'' and ''summation''. Thanks! [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 16:16, 29 May 2018 (UTC)

:I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using, ''This [[formal power series]] is the generating function.'' --[[User:Wcherowi|Bill Cherowitzo ]] ([[User talk:Wcherowi|talk]]) 17:04, 29 May 2018 (UTC)

:: Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is [https://en.wikipedia.org/w/index.php?title=Generating_function&diff=824616355&oldid=814318669 this one], which did not "restore" anything. The phrase "the summation of a series" makes no sense; if it did make sense, it would mean exactly the same as "the sum of the series". Indeed, the series ''is'' the sum; this sum is not a [real or complex] number because the individual summands are not [real or complex] numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:10, 29 May 2018 (UTC)
:::Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 19:31, 29 May 2018 (UTC)

:::: Your comment describes me has having "twice restored" something. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 29 May 2018 (UTC)

:::::Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry. [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 20:38, 29 May 2018 (UTC)
== Function* listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px|left]]
An editor has asked for a discussion to address the redirect [[Function*]]. Please participate in [[Wikipedia:Redirects for discussion/Log/2019 May 11#Function*|the redirect discussion]] if you wish to do so.  [[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 07:51, 11 May 2019 (UTC)

== Article is a mess ==

This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
* By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for Lambert, Bell, and formal Dirichlet series). The current version is trying to do ''way'' too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
* The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
* There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments".... [[Special:Contributions/2607:F720:F00:4834:E985:5B77:A9AF:D672|2607:F720:F00:4834:E985:5B77:A9AF:D672]] ([[User talk:2607:F720:F00:4834:E985:5B77:A9AF:D672|talk]]) 15:31, 15 May 2019 (UTC)

== must it be infinite? ==

Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial <math>(\frac{2}{5}x^1 + \frac{1}{5}x^2 + \frac{1}{5}x^3 + \frac{1}{5}x^4)^{64} </math>. Is that not a generating function because it's not infinite? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 14:56, 16 October 2019 (UTC)
: Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 18:17, 16 October 2019 (UTC)
:: The wiki-linking in the lede is also rather [[WP:SUBMARINE|submarine]]. It links to [[formal power series]] with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to [[formal power series]] ''again'' with the text "formal series". Next we get {{tq|Generating functions were first introduced by Abraham de Moivre in 1730}} — fine — {{tq| in order to solve the general linear recurrence problem.}} Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence problem just the challenge of understanding linear recurrences in general? [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 05:21, 17 October 2019 (UTC)

== Formula for generating function for a linear recursive sequrnce. ==

The following formula is really easy to use. Shall it be included in this article?

Let <math>s_n</math> be a linear recursive sequence of order k with initial conditions
<math> \{s_0, s_1, \ldots, s_{k-1}\}</math> and recursive relation
<math>s_n = \sum_{i=1}^k a_i s_{n-i}.</math>

Then the generating function for $s_n$ is given by the formula

<math>(\sum_{i=0}^{k-1} ( \sum_{j=0}^{i} (-a_j)* s_{i-j}) * x^{i-k})/f(x^{-1})</math> — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Kaiwang45|Kaiwang45]] ([[User talk:Kaiwang45#top|talk]] • [[Special:Contributions/Kaiwang45|contribs]]) 15:49, 27 July 2020 (UTC) 

== Blackboard bold formatting ==

{{reply|Quantling}} Greetings! Regarding [https://en.wikipedia.org/w/index.php?title=Generating_function&oldid=prev&diff=1146738276&diffmode=source this revert]...the use of {{tag|math}} is required by [[MOS:BBB]]. If we want the nearby markup to be consistent, that's fine; we would just need to convert it to also use {{tag|math}}. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 16:21, 27 March 2023 (UTC)
:{{reply to|Beland}} Good point. To be more consistent with [[MOS:STYLERET]], other possibilities are to use
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'', ''b'' ∈ '''N'''}}, {{math|''a'' ≥ 2}}, and {{math|0 ≤ ''b'' < ''a''}}.
:#In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}} and {{math|0 ≤ ''b'' < ''a''}} are integers.
:What do you think? —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:39, 27 March 2023 (UTC)
::{{reply|Quantling}} "{{math|''a''}} and {{math|''b''}} are integers" is certainly a lot less jargony than using the blackboard bold notation. -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 17:45, 27 March 2023 (UTC)
:::I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —[[User:Quantling|Quantling]] ([[User talk:Quantling|talk]] | [[Special:Contributions/Quantling|contribs]]) 17:55, 27 March 2023 (UTC)
::::Done; thanks for your help ironing this out! -- [[User:Beland|Beland]] ([[User talk:Beland|talk]]) 22:09, 27 March 2023 (UTC)

== Remove Sections ==

It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?

For starters, we should probably remove P-holonomic functions and J-representations and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.

Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! [[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 03:27, 25 November 2023 (UTC)

User talk:Yeetcode

2023-11-25T03:15:25Z

Yeetcode: /* General discussion on the mathematics situation in Wikipedia */ new section

== General discussion on the mathematics situation in Wikipedia ==

I would love it if people can bring up their concerns with specific Wikipedia mathematics articles and low level details on what a good fix would be, so that we can go ahead with it! [[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode#top|talk]]) 03:15, 25 November 2023 (UTC)

User:Yeetcode

2023-11-25T03:13:34Z

Yeetcode:

Mathematics enthusiast, and a firm believer in the right to free education.

User:Yeetcode

2023-11-25T03:12:13Z

Yeetcode: Undid revision 1186730256 by Yeetcode (talk)

Interested in basically everything. I specialise in discrete mathematics, though.

User:Yeetcode

2023-11-25T03:08:09Z

Yeetcode: ←Replaced content with 'Hi'

User:Yeetcode

2023-11-25T03:06:38Z

Yeetcode: ←Created page with 'Interested in basically everything. I specialise in discrete mathematics, though.'

Interested in basically everything. I specialise in discrete mathematics, though.

Generating function

2023-11-24T06:06:00Z

Yeetcode: /* Generating functions prove congruences */ The continued fraction was simply represented incorrectly in the typography. Fixed now.

{{Short description|Formal power series; coefficients encode information about a sequence indexed by natural numbers}}
{{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}
{{Very long|date=July 2022}}

In [[mathematics]], a '''generating function''' is a way of encoding an [[infinite sequence]] of numbers ({{math|''a''''n''}}) by treating them as the [[coefficient]]s of a [[formal power series]]. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' [[power series]] is not required to [[Convergent series|converge]]: in fact, the generating function is not actually regarded as a [[Function (mathematics)|function]], and the "variable" remains an [[Indeterminate (variable)|indeterminate]]. Generating functions were first introduced by [[Abraham de Moivre]] in 1730, in order to solve the general linear recurrence problem.<ref>{{cite book |author-link=Donald Knuth |first=Donald E. |last=Knuth |series=[[The Art of Computer Programming]] |volume=1 |title=Fundamental Algorithms |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |year=1997 |chapter=§1.2.9 Generating Functions}}</ref> One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

There are various types of generating functions, including '''ordinary generating functions''', '''exponential generating functions''', '''Lambert series''', '''Bell series''', and '''Dirichlet series'''; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in [[Closed-form expression|closed form]] (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate {{mvar|x}} may involve arithmetic operations, differentiation with respect to {{mvar|x}} and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of {{mvar|x}}. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of {{mvar|x}}, and which has the formal series as its [[series expansion]]; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a [[convergent series]] when a nonzero numeric value is substituted for {{mvar|x}}. Also, not all expressions that are meaningful as functions of {{mvar|x}} are meaningful as expressions designating formal series; for example, negative and fractional powers of {{mvar|x}} are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]. Generating functions are sometimes called '''generating series''',<ref>This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", ''[[Canadian Journal of Mathematics]]'' 3, [https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407 p. 405–411], but its use is rare before the year 2000; since then it appears to be increasing.</ref> in that a series of terms can be said to be the generator of its sequence of term coefficients.

==Definitions==

{{block quote
| text = ''A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.''
| author = [[George Pólya]]
| source = ''[[Mathematics and plausible reasoning]]'' (1954) }}

{{block quote
| text = ''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
| author = [[Herbert Wilf]]
| source = ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)}}

===Ordinary generating function (OGF)===

The ''ordinary generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">G(a_n;x)=\sum_{n=0}^\infty a_n x^n.</math>

When the term ''generating function'' is used without qualification, it is usually taken to mean an ordinary generating function.

If {{math|''a''''n''}} is the [[probability mass function]] of a [[discrete random variable]], then its ordinary generating function is called a [[probability-generating function]].

The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array {{math|''a''''m'',''n''}} (where {{mvar|n}} and {{mvar|m}} are natural numbers) is

<math display="block">G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n} x^m y^n.</math>

===Exponential generating function (EGF)===

The ''exponential generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.</math>

Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{harvnb|Flajolet|Sedgewick|2009|p=95}}</ref>

Another benefit of exponential generating functions is that they are useful in transferring linear [[recurrence relations]] to the realm of [[differential equations]]. For example, take the [[Fibonacci sequence]] {{math|{''fn''}<nowiki/>}} that satisfies the linear recurrence relation {{math|''f''''n''+2 {{=}} ''f''''n''+1 + ''f''''n''}}. The corresponding exponential generating function has the form

<math display="block">\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n</math>

and its derivatives can readily be shown to satisfy the differential equation {{math|EF{{pprime}}(''x'') {{=}} EF{{prime}}(''x'') + EF(''x'')}} as a direct analogue with the recurrence relation above. In this view, the factorial term {{math|''n''!}} is merely a counter-term to normalise the derivative operator acting on {{math|''x''''n''}}.

===Poisson generating function===
The ''Poisson generating function'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{PG}(a_n;x)=\sum _{n=0}^\infty a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).</math>

===Lambert series===
{{main article|Lambert series}}
The ''Lambert series'' of a sequence {{math|''a''''n''}} is

<math display="block">\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.</math>

The Lambert series coefficients in the power series expansions

<math display="block">b_n := [x^n] \operatorname{LG}(a_n;x)</math>

for integers {{math|''n'' ≥ 1}} are related by the [[divisor sum]]

<math display="block">b_n = \sum_{d|n} a_d.</math>

The main article provides several more classical, or at least well-known examples related to special [[arithmetic functions]] in [[number theory]].

In a Lambert series the index {{mvar|n}} starts at 1, not at 0, as the first term would otherwise be undefined.

===Bell series===

The [[Bell series]] of a sequence {{math|''a''''n''}} is an expression in terms of both an indeterminate {{mvar|x}} and a prime {{mvar|p}} and is given by<ref>{{Apostol IANT}} pp.42–43</ref>

<math display="block">\operatorname{BG}_p(a_n;x) = \sum_{n=0}^\infty a_{p^n}x^n.</math>

===Dirichlet series generating functions (DGFs)===

[[Formal Dirichlet series]] are often classified as generating functions, although they are not strictly formal power series. The ''Dirichlet series generating function'' of a sequence {{math|''a''''n''}} is<ref name=W56>{{harvnb|Wilf|1994|p=56}}</ref>

<math display="block">\operatorname{DG}(a_n;s)=\sum _{n=1}^\infty \frac{a_n}{n^s}.</math>

The Dirichlet series generating function is especially useful when {{math|''a''''n''}} is a [[multiplicative function]], in which case it has an [[Euler product]] expression<ref name=W59>{{harvnb|Wilf|1994|p=59}}</ref> in terms of the function's Bell series

<math display="block">\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.</math>

If {{math|''a''''n''}} is a [[Dirichlet character]] then its Dirichlet series generating function is called a [[Dirichlet L-series|Dirichlet {{mvar|L}}-series]]. We also have a relation between the pair of coefficients in the [[Lambert series]] expansions above and their DGFs. Namely, we can prove that

<math display="block">[x^n] \operatorname{LG}(a_n; x) = b_n</math>

if and only if

<math display="block">\operatorname{DG}(a_n;s) \zeta(s) = \operatorname{DG}(b_n;s),</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]].<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |last3=Heath-Brown |first3=D.R |last4=Silverman |first4=J.H. |title=An Introduction to the Theory of Numbers|url=https://archive.org/details/introductiontoth00ghha_922|url-access=limited|publisher=Oxford University Press |page=[https://archive.org/details/introductiontoth00ghha_922/page/n357 339]|edition=6th |isbn=9780199219858 |year=2008}}</ref>

===Polynomial sequence generating functions===

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of [[binomial type]] are generated by

<math display="block">e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n</math>

where {{math|''p''''n''(''x'')}} is a sequence of polynomials and {{math|''f''(''t'')}} is a function of a certain form. [[Sheffer sequence]]s are generated in a similar way. See the main article [[generalized Appell polynomials]] for more information.

== Ordinary generating functions ==

=== Examples of generating functions for simple sequences ===

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the [[Poincaré polynomial]] and others.

A fundamental generating function is that of the constant sequence {{nowrap|1, 1, 1, 1, 1, 1, 1, 1, 1, ...}}, whose ordinary generating function is the [[Geometric_series#Closed-form_formula|geometric series]]

<math display="block">\sum_{n=0}^\infty x^n= \frac{1}{1-x}.</math>

The left-hand side is the [[Maclaurin series]] expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by {{math|1 − ''x''}}, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of {{math|''x''0}} are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the [[multiplicative inverse]] of {{math|1 − ''x''}} in the ring of power series.

Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution {{math|''x'' → ''ax''}} gives the generating function for the [[Geometric progression|geometric sequence]] {{math|1, ''a'', ''a''2, ''a''3, ...}} for any constant {{mvar|a}}:

<math display="block">\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.</math>

(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,

<math display="block">\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.</math>

One can also introduce regular gaps in the sequence by replacing {{mvar|x}} by some power of {{mvar|x}}, so for instance for the sequence {{nowrap|1, 0, 1, 0, 1, 0, 1, 0, ...}} (which skips over {{math|''x'', ''x''3, ''x''5, ...}}) one gets the generating function

<math display="block">\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.</math>

By squaring the initial generating function, or by finding the derivative of both sides with respect to {{mvar|x}} and making a change of running variable {{math|''n'' → ''n'' + 1}}, one sees that the coefficients form the sequence {{nowrap|1, 2, 3, 4, 5, ...}}, so one has

<math display="block">\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},</math>

and the third power has as coefficients the [[triangular number]]s {{nowrap|1, 3, 6, 10, 15, 21, ...}} whose term {{mvar|n}} is the [[binomial coefficient]] {{math|{{pars|s=150%|{{su|p=''n'' + 2|b=2|a=c}}}}}}, so that

<math display="block">\sum_{n=0}^\infty\binom{n+2}2 x^n= \frac{1}{(1-x)^3}.</math>

More generally, for any non-negative integer {{mvar|k}} and non-zero real value {{mvar|a}}, it is true that

<math display="block">\sum_{n=0}^\infty a^n\binom{n+k}k x^n= \frac{1}{(1-ax)^{k+1}}\,.</math>

Since

<math display="block">2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0 = 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2,</math>

one can find the ordinary generating function for the sequence {{nowrap|0, 1, 4, 9, 16, ...}} of [[square number]]s by linear combination of binomial-coefficient generating sequences:

<math display="block">G(n^2;x) = \sum_{n=0}^\infty n^2x^n = \frac{2}{(1-x)^3} - \frac{3}{(1-x)^2} + \frac{1}{1-x} = \frac{x(x+1)}{(1-x)^3}.</math>

We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the [[geometric series]] in the following form:

<math display="block">\begin{align}
G(n^2;x)
& = \sum_{n=0}^\infty n^2x^n \\[4px]
& = \sum_{n=0}^\infty n(n-1) x^n + \sum_{n=0}^\infty n x^n \\[4px]
& = x^2 D^2\left[\frac{1}{1-x}\right] + x D\left[\frac{1}{1-x}\right] \\[4px]
& = \frac{2 x^2}{(1-x)^3} + \frac{x}{(1-x)^2} =\frac{x(x+1)}{(1-x)^3}.
\end{align}</math>

By induction, we can similarly show for positive integers {{math|''m'' ≥ 1}} that<ref>{{cite journal|first1= Michael Z. | last1=Spivey | title=Combinatorial Sums and Finite Differences | year=2007 |journal = Discrete Math. |doi = 10.1016/j.disc.2007.03.052 | volume=307|number=24|pages=3130–3146|mr=2370116|doi-access=free }}</ref><ref>{{cite arXiv|first1=R. J. |last1=Mathar|year=2012|eprint=1207.5845|title=Yet another table of integrals|class=math.CA}} v4 eq. (0.4)</ref>

<math display="block">n^m = \sum_{j=0}^m \begin{Bmatrix} m \\ j \end{Bmatrix} \frac{n!}{(n-j)!}, </math>

where {{math|{{resize|150%|{}}{{su|p=''n''|b=''k''}}{{resize|150%|}<nowiki/>}}}} denote the [[Stirling numbers of the second kind]] and where the generating function

<math display="block">\sum_{n = 0}^\infty \frac{n!}{ (n-j)!} \, z^n = \frac{j! \cdot z^j}{(1-z)^{j+1}},</math>

so that we can form the analogous generating functions over the integral {{mvar|m}}th powers generalizing the result in the square case above. In particular, since we can write

<math display="block">\frac{z^k}{(1-z)^{k+1}} = \sum_{i=0}^k \binom{k}{i} \frac{(-1)^{k-i}}{(1-z)^{i+1}},</math>

we can apply a well-known finite sum identity involving the [[Stirling numbers]] to obtain that<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Table 265 in §6.1}} for finite sum identities involving the Stirling number triangles.</ref>

<math display="block">\sum_{n = 0}^\infty n^m z^n = \sum_{j=0}^m \begin{Bmatrix} m+1 \\ j+1 \end{Bmatrix} \frac{(-1)^{m-j} j!}{(1-z)^{j+1}}. </math>

=== Rational functions ===
{{Main|Linear recursive sequence}}
The ordinary generating function of a sequence can be expressed as a [[rational function]] (the ratio of two finite-degree polynomials) if and only if the sequence is a [[linear recursive sequence]] with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear [[finite difference equation]] with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive [[Binet's formula]] for the [[Fibonacci numbers]] via generating function techniques.

We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate ''quasi-polynomial'' sequences of the form <ref name="GFLECT">{{harvnb|Lando|2003|loc=§2.4}}</ref>

<math display="block">f_n = p_1(n) \rho_1^n + \cdots + p_\ell(n) \rho_\ell^n, </math>

where the reciprocal roots, <math>\rho_i \isin \mathbb{C}</math>, are fixed scalars and where {{math|''p''''i''(''n'')}} is a polynomial in {{mvar|n}} for all {{math|1 ≤ ''i'' ≤ ''ℓ''}}.

In general, [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard products]] of rational functions produce rational generating functions. Similarly, if

<math display="block">F(s, t) := \sum_{m,n \geq 0} f(m, n) w^m z^n</math>

is a bivariate rational generating function, then its corresponding ''diagonal generating function'',

<math display="block">\operatorname{diag}(F) := \sum_{n = 0}^\infty f(n, n) z^n,</math>

is ''algebraic''. For example, if we let<ref>Example from {{cite book |chapter=§6.3 |first1=Richard P. |last1=Stanley |first2=Sergey |last2=Fomin |title=Enumerative Combinatorics: Volume 2 |url=https://books.google.com/books?id=zg5wDqT6T-UC |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-78987-5 |series=Cambridge Studies in Advanced Mathematics |volume=62}}</ref>

<math display="block">F(s, t) := \sum_{i,j \geq 0} \binom{i+j}{i} s^i t^j = \frac{1}{1-s-t}, </math>

then this generating function's diagonal coefficient generating function is given by the well-known OGF formula

<math display="block">\operatorname{diag}(F) = \sum_{n = 0}^\infty \binom{2n}{n} z^n = \frac{1}{\sqrt{1-4z}}. </math>

This result is computed in many ways, including [[Cauchy's integral formula]] or [[contour integration]], taking complex [[residue (complex analysis)|residue]]s, or by direct manipulations of [[formal power series]] in two variables.

=== Operations on generating functions ===

==== Multiplication yields convolution ====
{{Main|Cauchy product}}
Multiplication of ordinary generating functions yields a discrete [[convolution]] (the [[Cauchy product]]) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general [[Euler–Maclaurin formula]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''an''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[Generating function#Convolution (Cauchy products)|section on convolutions]] in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.

==== Shifting sequence indices ====

For integers {{math|''m'' ≥ 1}}, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of {{math|⟨ ''g''''n'' − ''m'' ⟩}} and {{math|⟨ ''g''''n'' + ''m'' ⟩}}, respectively:

<math display="block">\begin{align}
& z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n \\[4px]
& \frac{G(z) - g_0 - g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m} = \sum_{n = 0}^\infty g_{n+m} z^n.
\end{align}</math>

==== Differentiation and integration of generating functions ====

We have the following respective power series expansions for the first derivative of a generating function and its integral:

<math display="block">\begin{align}
G'(z) & = \sum_{n = 0}^\infty (n+1) g_{n+1} z^n \\[4px]
z \cdot G'(z) & = \sum_{n = 0}^\infty n g_{n} z^n \\[4px]
\int_0^z G(t) \, dt & = \sum_{n = 1}^\infty \frac{g_{n-1}}{n} z^n.
\end{align}</math>

The differentiation–multiplication operation of the second identity can be repeated {{mvar|k}} times to multiply the sequence by {{math|''n''''k''}}, but that requires alternating between differentiation and multiplication. If instead doing {{mvar|k}} differentiations in sequence, the effect is to multiply by the {{mvar|k}}th [[falling factorial]]:

<math display="block"> z^k G^{(k)}(z) = \sum_{n = 0}^\infty n^\underline{k} g_n z^n = \sum_{n = 0}^\infty n (n-1) \dotsb (n-k+1) g_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

Using the [[Stirling numbers of the second kind]], that can be turned into another formula for multiplying by <math>n^k</math> as follows (see the main article on [[Generating function transformation#Derivative transformations|generating function transformations]]):

<math display="block"> \sum_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} z^j F^{(j)}(z) = \sum_{n = 0}^\infty n^k f_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the [[Generating function transformation#Derivative transformations|zeta series transformation]] and its generalizations defined as a derivative-based [[generating function transformation|transformation of generating functions]], or alternately termwise by and performing an [[Generating function transformation#Polylogarithm series transformations|integral transformation]] on the sequence generating function. Related operations of performing [[fractional calculus|fractional integration]] on a sequence generating function are discussed [[Generating function transformation#Fractional integrals and derivatives|here]].

==== Enumerating arithmetic progressions of sequences ====
In this section we give formulas for generating functions enumerating the sequence {{math|{''f''''an'' + ''b''}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}}, {{math|0 ≤ ''b'' < ''a''}}, and {{math|''a''}} and {{math|''b''}} are integers (see the [[generating function transformation|main article on transformations]]). For {{math|''a'' {{=}} 2}}, this is simply the familiar decomposition of a function into [[even and odd functions|even and odd parts]] (i.e., even and odd powers):

<math display="block">\begin{align}
\sum_{n = 0}^\infty f_{2n} z^{2n} &= \frac{F(z) + F(-z)}{2} \\[4px]
\sum_{n = 0}^\infty f_{2n+1} z^{2n+1} &= \frac{F(z) - F(-z)}{2}.
\end{align}</math>

More generally, suppose that {{math|''a'' ≥ 3}} and that {{math|''ωa'' {{=}} exp {{sfrac|2''πi''|''a''}}}} denotes the {{mvar|a}}th [[root of unity|primitive root of unity]]. Then, as an application of the [[discrete Fourier transform]], we have the formula<ref name="TAOCPV1">{{harvnb|Knuth|1997|loc=§1.2.9}}</ref>

<math display="block">\sum_{n = 0}^\infty f_{an+b} z^{an+b} = \frac{1}{a} \sum_{m=0}^{a-1} \omega_a^{-mb} F\left(\omega_a^m z\right).</math>

For integers {{math|''m'' ≥ 1}}, another useful formula providing somewhat ''reversed'' floored arithmetic progressions — effectively repeating each coefficient {{mvar|m}} times — are generated by the identity<ref>Solution to {{harvnb|Graham|Knuth|Patashnik|1994|p=569, exercise 7.36}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{\left\lfloor \frac{n}{m} \right\rfloor} z^n = \frac{1-z^m}{1-z} F(z^m) = \left(1 + z + \cdots + z^{m-2} + z^{m-1}\right) F(z^m).</math>

==={{math|''P''}}-recursive sequences and holonomic generating functions===

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>

<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''ci''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> spanned by the set of all of its derivatives is finite dimensional.

Since we can clear denominators if need be in the previous equation, we may assume that the functions, {{math|''ci''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form

<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''0}} and where the {{math|''ĉi''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.

====Examples====

The functions {{math|''e''''z''}}, {{math|log ''z''}}, {{math|cos ''z''}}, {{math|arcsin ''z''}}, <math>\sqrt{1 + z}</math>, the [[dilogarithm]] function {{math|Li2(''z'')}}, the [[generalized hypergeometric function]]s {{math|''pFq''(...; ...; ''z'')}} and the functions defined by the power series

<math display="block">\sum_{n = 0}^\infty \frac{z^n}{(n!)^2}</math>

and the non-convergent

<math display="block">\sum_{n = 0}^\infty n! \cdot z^n</math>

are all holonomic.

Examples of {{mvar|P}}-recursive sequences with holonomic generating functions include {{math|''f''''n'' ≔ {{sfrac|1|''n'' + 1}} {{pars|s=150%|{{su|p=2''n''|b=''n''|a=c}}}}}} and {{math|''f''''n'' ≔ {{sfrac|2''n''|''n''2 + 1}}}}, where sequences such as <math>\sqrt{n}</math> and {{math|log ''n''}} are ''not'' {{mvar|P}}-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as {{math|tan ''z''}}, {{math|sec ''z''}}, and [[Gamma function|{{math|Γ(''z'')}}]] are ''not'' holonomic functions.

====Software for working with ''{{mvar|P}}''-recursive sequences and holonomic generating functions====

Tools for processing and working with {{mvar|P}}-recursive sequences in ''[[Mathematica]]'' include the software packages provided for non-commercial use on the [https://www.risc.jku.at/research/combinat/software/ RISC Combinatorics Group algorithmic combinatorics software] site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>'''Guess'''</code> package for guessing ''{{mvar|P}}-recurrences'' for arbitrary input sequences (useful for [[experimental mathematics]] and exploration) and the <code>'''Sigma'''</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to {{mvar|P}}-recurrences involving generalized [[harmonic number]]s.<ref>{{cite journal|last1=Schneider|first1=C.|title=Symbolic Summation Assists Combinatorics|journal=Sem. Lothar. Combin.|date=2007|volume=56|pages=1–36 |url=http://www.emis.de/journals/SLC/wpapers/s56schneider.html}}</ref> Other packages listed on this particular RISC site are targeted at working with holonomic ''generating functions'' specifically.


=== Relation to discrete-time Fourier transform ===
{{Main|Discrete-time Fourier transform}}
When the series [[Absolute convergence|converges absolutely]],
<math display="block">G \left ( a_n; e^{-i \omega} \right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math>
is the discrete-time Fourier transform of the sequence {{math|''a''0, ''a''1, ...}}.

=== Asymptotic growth of a sequence ===
In calculus, often the growth rate of the coefficients of a power series can be used to deduce a [[radius of convergence]] for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the [[Asymptotic analysis|asymptotic growth]] of the underlying sequence.

For instance, if an ordinary generating function {{math|''G''(''a''''n''; ''x'')}} that has a finite radius of convergence of {{mvar|r}} can be written as

<math display="block">G(a_n; x) = \frac{A(x) + B(x) \left (1- \frac{x}{r} \right )^{-\beta}}{x^\alpha}</math>

where each of {{math|''A''(''x'')}} and {{math|''B''(''x'')}} is a function that is [[analytic function|analytic]] to a radius of convergence greater than {{mvar|r}} (or is [[Entire function|entire]]), and where {{math|''B''(''r'') ≠ 0}} then

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1}\left(\frac{1}{r}\right)^n \sim \frac{B(r)}{r^{\alpha}} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n = \frac{B(r)}{r^\alpha} \left(\!\!\binom{\beta}{n}\!\!\right)\left(\frac{1}{r}\right)^n\,,</math>
using the [[gamma function]], a [[binomial coefficient]], or a [[multiset coefficient]].

Often this approach can be iterated to generate several terms in an asymptotic series for {{math|''a''''n''}}. In particular,

<math display="block">G\left(a_n - \frac{B(r)}{r^\alpha} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n; x \right) = G(a_n; x) - \frac{B(r)}{r^\alpha} \left(1 - \frac{x}{r}\right)^{-\beta}\,.</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of {{mvar|A}}, {{mvar|B}}, {{mvar|α}}, {{mvar|β}}, and {{mvar|r}} to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is {{math|{{sfrac|''a''''n''|''n''!}}}} that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

==== Asymptotic growth of the sequence of squares ====
As derived above, the ordinary generating function for the sequence of squares is

<math display="block">G(n^2; x) = \frac{x(x+1)}{(1-x)^3}.</math>

With {{math|1=''r'' = 1}}, {{math|1=''α'' = −1}}, {{math|1=''β'' = 3}}, {{math|1=''A''(''x'') = 0}}, and {{math|1=''B''(''x'') = ''x'' + 1}}, we can verify that the squares grow as expected, like the squares:

<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left (\frac{1}{r} \right)^n = \frac{1+1}{1^{-1}\,\Gamma(3)}\,n^{3-1} \left(\frac1 1\right)^n = n^2.</math>

==== Asymptotic growth of the Catalan numbers ====
{{Main|Catalan number}}

The ordinary generating function for the [[Catalan number]]s is

<math display="block">G(C_n; x) = \frac{1-\sqrt{1-4x}}{2x}.</math>

With {{math|1=''r'' = {{sfrac|1|4}}}}, {{math|1=''α'' = 1}}, {{math|1=''β'' = −{{sfrac|1|2}}}}, {{math|1=''A''(''x'') = {{sfrac|1|2}}}}, and {{math|1=''B''(''x'') = −{{sfrac|1|2}}}}, we can conclude that, for the Catalan numbers,

<math display="block">C_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left(\frac{1}{r} \right)^n = \frac{-\frac12}{\left(\frac14\right)^1 \Gamma\left(-\frac12\right)} \, n^{-\frac12-1} \left(\frac{1}{\,\frac14\,}\right)^n = \frac{4^n}{n^\frac32 \sqrt\pi}.</math>

=== Bivariate and multivariate generating functions ===
One can define generating functions in several variables for arrays with several indices. These are called '''multivariate generating functions''' or, sometimes, '''super generating functions'''. For two variables, these are often called '''bivariate generating functions'''.

For instance, since {{math|(1 + ''x'')''n''}} is the ordinary generating function for [[binomial coefficients]] for a fixed {{mvar|n}}, one may ask for a bivariate generating function that generates the binomial coefficients {{math|{{pars|s=150%|{{su|p=''n''|b=''k''|a=c}}}}}} for all {{mvar|k}} and {{mvar|n}}. To do this, consider {{math|(1 + ''x'')''n''}} itself as a sequence in {{mvar|n}}, and find the generating function in {{mvar|y}} that has these sequence values as coefficients. Since the generating function for {{math|''a''''n''}} is

<math display="block">\frac{1}{1-ay},</math>

the generating function for the binomial coefficients is:

<math display="block">\sum_{n,k} \binom{n}{k} x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}.</math>

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-order accurate]] power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the [[Jacobi-type continued fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>

<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
& = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''jn'' ≔ [''zn''] ''J''[∞](''z'')}}, in the previous equations correspond to matrix solutions of the equations

<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
\begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
\begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''0 ≡ ''k''0,0 {{=}} 1}}, {{math|''jn'' {{=}} ''k''0,''n''}} for {{math|''n'' ≥ 1}}, {{math|''k''''r'',''s'' {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by

<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''[∞](''z'')}}, expanded by

<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''Ph''(''z'')}} and {{math|''Qh''(''z'')}}, defined recursively by

<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv''h''(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''jn''}}, ''and'' for {{math|''Mh'' ≔ ab2 ⋯ ab''h'' + 1}} if {{math|''h'' ‖ ''M''''h''}} then we have the congruence

<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab''i''}<nowiki/>}} and {{math|{''c''''i''}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.

==== Examples ====

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''jn''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

:{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.

==Examples==

Generating functions for the sequence of [[square number]]s {{math|''a''''n'' {{=}} ''n''2}} are:

===Ordinary generating function===
<math display="block">G(n^2;x)=\sum_{n=0}^\infty n^2x^n = \frac{x(x+1)}{(1-x)^3}</math>

===Exponential generating function===
<math display="block">\operatorname{EG}(n^2;x)=\sum _{n=0}^\infty \frac{n^2x^n}{n!}=x(x+1)e^x</math>

===Lambert series===

As an example of a Lambert series identity not given in the [[Lambert series|main article]], we can show that for {{math|{{abs|''x''}}, {{abs|''xq''}} < 1}} we have that <ref>{{cite web|title=Lambert series identity|url=https://mathoverflow.net/q/140418 |website=Math Overflow|date=2017}}</ref>

<math display="block">\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n}, </math>

where we have the special case identity for the generating function of the [[divisor function]], {{math|''d''(''n'') ≡ ''σ''0(''n'')}}, given by

<math display="block">\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}. </math>

===Bell series===
<math display="block">\operatorname{BG}_p\left(n^2;x\right)=\sum_{n=0}^\infty \left(p^{n}\right)^2x^n=\frac{1}{1-p^2x}</math>

===Dirichlet series generating function===
<math display="block">\operatorname{DG}\left(n^2;s\right)=\sum_{n=1}^\infty \frac{n^2}{n^s}=\zeta(s-2),</math>

using the [[Riemann zeta function]].

The sequence {{mvar|ak}} generated by a [[Dirichlet series]] generating function (DGF) corresponding to:

<math display="block">\operatorname{DG}(a_k;s)=\zeta(s)^m</math>

where {{math|''ζ''(''s'')}} is the [[Riemann zeta function]], has the ordinary generating function:

<math display="block">\sum_{k=1}^{k=n} a_k x^k = x + \binom{m}{1} \sum_{2 \leq a \leq n} x^{a} + \binom{m}{2}\underset{ab \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty} x^{ab} + \binom{m}{3}\underset{abc \leq n}{\sum_{a = 2}^\infty \sum_{c = 2}^\infty \sum_{b = 2}^\infty} x^{abc} + \binom{m}{4}\underset{abcd \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty \sum_{c = 2}^\infty \sum_{d = 2}^\infty} x^{abcd} + \cdots</math>

===Multivariate generating functions===
Multivariate generating functions arise in practice when calculating the number of [[contingency tables]] of non-negative integers with specified row and column totals. Suppose the table has {{mvar|r}} rows and {{mvar|c}} columns; the row sums are {{math|''t''1, ''t''2 ... ''tr''}} and the column sums are {{math|''s''1, ''s''2 ... ''sc''}}. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal| doi=10.1214/aos/1176343649| last=Good| first=I. J.| title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables| journal=[[Annals of Statistics]]| year=1986| volume=4| issue=6|pages=1159–1189| doi-access=free}}</ref> the number of such tables is the coefficient of

<math display="block">x_1^{t_1}\cdots x_r^{t_r}y_1^{s_1}\cdots y_c^{s_c}</math>

in

<math display="block">\prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.</math>

In the bivariate case, non-polynomial double sum examples of so-termed "''double''" or "''super''" generating functions of the form

<math display="block">G(w, z) := \sum_{m,n \geq 0} g_{m,n} w^m z^n</math>

include the following two-variable generating functions for the [[binomial coefficients]], the [[Stirling numbers]], and the [[Eulerian numbers]]:<ref>See the usage of these terms in {{harvnb|Graham|Knuth|Patashnik|1994|loc=§7.4}} on special sequence generating functions.</ref>

<math display="block">\begin{align}
e^{z+wz} & = \sum_{m,n \geq 0} \binom{n}{m} w^m \frac{z^n}{n!} \\[4px]
e^{w(e^z-1)} & = \sum_{m,n \geq 0} \begin{Bmatrix} n \\ m \end{Bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1}{(1-z)^w} & = \sum_{m,n \geq 0} \begin{bmatrix} n \\ m \end{bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1-w}{e^{(w-1)z}-w} & = \sum_{m,n \geq 0} \left\langle\begin{matrix} n \\ m \end{matrix} \right\rangle w^m \frac{z^n}{n!} \\[4px]
\frac{e^w-e^z}{w e^z-z e^w} &= \sum_{m,n \geq 0} \left\langle\begin{matrix} m+n+1 \\ m \end{matrix} \right\rangle \frac{w^m z^n}{(m+n+1)!}.
\end{align}</math>

==Applications==

===Various techniques: Evaluating sums and tackling other problems with generating functions===

====Example 1: A formula for sums of harmonic numbers====

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when {{math|''sn'' {{=}} Σ{{su|b=''k'' {{=}} 0|p=''n''}} ''ak''}}. We then know that {{math|''S''(''z'') {{=}} {{sfrac|''A''(''z'')|1 − ''z''}}}} for the corresponding ordinary generating functions.

For example, we can manipulate
<math display="block">s_n=\sum_{k=1}^{n} H_{k}\,,</math>
where {{math|''Hk'' {{=}} 1 + {{sfrac|1|2}} + ⋯ + {{sfrac|1|''k''}}}} are the [[harmonic number]]s. Let
<math display="block">H(z) = \sum_{n = 1}^\infty{H_n z^n}</math>
be the ordinary generating function of the harmonic numbers. Then
<math display="block">H(z) = \frac{1}{1-z}\sum_{n = 1}^\infty \frac{z^n}{n}\,,</math>
and thus
<math display="block">S(z) = \sum_{n = 1}^\infty{s_n z^n} = \frac{1}{(1-z)^2}\sum_{n = 1}^\infty \frac{z^n}{n}\,.</math>

Using
<math display="block">\frac{1}{(1-z)^2} = \sum_{n = 0}^\infty (n+1)z^n\,,</math>
[[Generating function#Convolution (Cauchy products)|convolution]] with the numerator yields
<math display="block">s_n = \sum_{k = 1}^{n} \frac{n+1-k}{k} = (n+1)H_n - n\,,</math>
which can also be written as
<math display="block">\sum_{k = 1}^{n}{H_k} = (n+1)(H_{n+1} - 1)\,.</math>

====Example 2: Modified binomial coefficient sums and the binomial transform====

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence {{math|⟨ ''fn'' ⟩}} we define the two sequences of sums
<math display="block">\begin{align}
s_n &:= \sum_{m=0}^n \binom{n}{m} f_m 3^{n-m} \\[4px]
\tilde{s}_n &:= \sum_{m=0}^n \binom{n}{m} (m+1)(m+2)(m+3) f_m 3^{n-m}\,,
\end{align}</math>
for all {{math|''n'' ≥ 0}}, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the [[binomial transform]] to write the generating function for the first sum as
<math display="block">S(z) = \frac{1}{1-3z} F\left(\frac{z}{1-3z}\right). </math>

Since the generating function for the sequence {{math|⟨ (''n'' + 1)(''n'' + 2)(''n'' + 3) ''fn'' ⟩}} is given by
<math display="block">6 F(z) + 18z F'(z) + 9z^2 F''(z) + z^3 F'''(z)</math>
we may write the generating function for the second sum defined above in the form
<math display="block">\tilde{S}(z) = \frac{6}{(1-3z)} F\left(\frac{z}{1-3z}\right)+\frac{18z}{(1-3z)^2} F'\left(\frac{z}{1-3z}\right)+\frac{9z^2}{(1-3z)^3} F''\left(\frac{z}{1-3z}\right)+\frac{z^3}{(1-3z)^4} F'''\left(\frac{z}{1-3z}\right). </math>

In particular, we may write this modified sum generating function in the form of
<math display="block">a(z) \cdot S(z) + b(z) \cdot z S'(z) + c(z) \cdot z^2 S''(z) + d(z) \cdot z^3 S'''(z), </math>
for {{math|''a''(''z'') {{=}} 6(1 − 3''z'')3}}, {{math|''b''(''z'') {{=}} 18(1 − 3''z'')3}}, {{math|''c''(''z'') {{=}} 9(1 − 3''z'')3}}, and {{math|''d''(''z'') {{=}} (1 − 3''z'')3}}, where {{math|(1 − 3''z'')3 {{=}} 1 − 9''z'' + 27''z''2 − 27''z''3}}.

Finally, it follows that we may express the second sums through the first sums in the following form:
<math display="block">\begin{align}
\tilde{s}_n & = [z^n]\left(6(1-3z)^3 \sum_{n = 0}^\infty s_n z^n + 18 (1-3z)^3 \sum_{n = 0}^\infty n s_n z^n + 9 (1-3z)^3 \sum_{n = 0}^\infty n(n-1) s_n z^n + (1-3z)^3 \sum_{n = 0}^\infty n(n-1)(n-2) s_n z^n\right) \\[4px]
& = (n+1)(n+2)(n+3) s_n - 9 n(n+1)(n+2) s_{n-1} + 27 (n-1)n(n+1) s_{n-2} - (n-2)(n-1)n s_{n-3}.
\end{align}</math>

====Example 3: Generating functions for mutually recursive sequences====

In this example, we reformulate a generating function example given in Section 7.3 of ''Concrete Mathematics'' (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted {{math|''Un''}}) to tile a 3-by-{{mvar|n}} rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, {{math|''Vn''}}, be defined as the number of ways to cover a 3-by-{{mvar|n}} rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a [[Closed-form expression|closed form]] formula for {{math|''Un''}} without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series

<math display="block">\begin{align}
U(z) = 1 + 3z^2 + 11 z^4 + 41 z^6 + \cdots, \\
V(z) = z + 4z^3 + 15 z^5 + 56 z^7 + \cdots.
\end{align}</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-{{mvar|n}} rectangle, we are able to express the following mutually dependent, or ''mutually recursive'', recurrence relations for our two sequences when {{math|''n'' ≥ 2}} defined as above where {{math|''U''0 {{=}} 1}}, {{math|''U''1 {{=}} 0}}, {{math|''V''0 {{=}} 0}}, and {{math|''V''1 {{=}} 1}}:
<math display="block">\begin{align}
U_n & = 2 V_{n-1} + U_{n-2} \\
V_n & = U_{n-1} + V_{n-2}.
\end{align}</math>

Since we have that for all integers {{math|''m'' ≥ 0}}, the index-shifted generating functions satisfy{{noteTag|Incidentally, we also have a corresponding formula when {{math|''m'' < 0}} given by
<math display="block">\sum_{n = 0}^\infty g_{n+m} z^n = \frac{G(z) - g_0 -g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m}\,.</math>}}
<math display="block">z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n\,,</math>
we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
<math display="block">\begin{align}
U(z) & = 2z V(z) + z^2 U(z) + 1 \\
V(z) & = z U(z) + z^2 V(z) = \frac{z}{1-z^2} U(z),
\end{align}</math>
which then implies by solving the system of equations (and this is the particular trick to our method here) that
<math display="block">U(z) = \frac{1-z^2}{1-4z^2+z^4} = \frac{1}{3-\sqrt{3}} \cdot \frac{1}{1-\left(2+\sqrt{3}\right) z^2} + \frac{1}{3 + \sqrt{3}} \cdot \frac{1}{1-\left(2-\sqrt{3}\right) z^2}. </math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that {{math|''U''2''n'' + 1 ≡ 0}} and that
<math display="block">U_{2n} = \left\lceil \frac{\left(2+\sqrt{3}\right)^n}{3-\sqrt{3}} \right\rceil\,, </math>
for all integers {{math|''n'' ≥ 0}}. We also note that the same shifted generating function technique applied to the second-order [[recurrence relation|recurrence]] for the [[Fibonacci numbers]] is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on [[rational functions]] given above.

===Convolution (Cauchy products)===

A discrete ''convolution'' of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see [[Cauchy product]]).

#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are ordinary generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow [z^n]C(z) = \sum_{k=0}^{n}{a_k b_{n-k}}</math>
#Consider {{math|''A''(''z'')}} and {{math|''B''(''z'')}} are exponential generating functions. <math display="block">C(z) = A(z)B(z) \Leftrightarrow \left[\frac{z^n}{n!}\right]C(z) = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}</math>
#Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block">C(z) = F(z) G(z) H(z) \Leftrightarrow [z^n]C(z) = \sum_{j+k+ l=n} f_j g_k h_ l</math>
#Consider the {{mvar|m}}-fold convolution of a sequence with itself for some positive integer {{math|''m'' ≥ 1}} (see the example below for an application) <math display="block">C(z) = G(z)^m \Leftrightarrow [z^n]C(z) = \sum_{k_1+k_2+\cdots+k_m=n} g_{k_1} g_{k_2} \cdots g_{k_m}</math>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the [[probability generating function]], or ''pgf'', of a random variable {{mvar|Z}} is denoted by {{math|''GZ''(''z'')}}, then we can show that for any two random variables <ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=§8.3}}</ref>
<math display="block">G_{X+Y}(z) = G_X(z) G_Y(z)\,, </math>
if {{mvar|X}} and {{mvar|Y}} are independent. Similarly, the number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
<math display="block">C(z) = \frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}, </math>
and moreover, if we allow the {{mvar|n}} cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the [[partition function (mathematics)|partition function]] generating function expanded by the infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product of
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>

====Example: The generating function for the Catalan numbers====

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the [[Catalan numbers]], {{math|''Cn''}}. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product {{math|''x''0 · ''x''1 ·⋯· ''xn''}} so that the order of multiplication is completely specified. For example, {{math|''C''2 {{=}} 2}} which corresponds to the two expressions {{math|''x''0 · (''x''1 · ''x''2)}} and {{math|(''x''0 · ''x''1) · ''x''2}}. It follows that the sequence satisfies a recurrence relation given by
<math display="block">C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k} + \delta_{n,0} = C_0 C_{n-1} + C_1 C_{n-2} + \cdots + C_{n-1} C_0 + \delta_{n,0}\,,\quad n \geq 0\,, </math>
and so has a corresponding convolved generating function, {{math|''C''(''z'')}}, satisfying
<math display="block">C(z) = z \cdot C(z)^2 + 1\,.</math>

Since {{math|''C''(0) {{=}} 1 ≠ ∞}}, we then arrive at a formula for this generating function given by
<math display="block">C(z) = \frac{1-\sqrt{1-4z}}{2z} = \sum_{n = 0}^\infty \frac{1}{n+1}\binom{2n}{n} z^n\,.</math>

Note that the first equation implicitly defining {{math|''C''(''z'')}} above implies that
<math display="block">C(z) = \frac{1}{1-z \cdot C(z)} \,, </math>
which then leads to another "simple" (of form) continued fraction expansion of this generating function.

====Example: Spanning trees of fans and convolutions of convolutions====

A ''fan of order {{mvar|n}}'' is defined to be a graph on the vertices {{math|{0, 1, ..., ''n''}<nowiki/>}} with {{math|2''n'' − 1}} edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other {{mvar|n}} vertices, and vertex <math>k</math> is connected by a single edge to the next vertex {{math|''k'' + 1}} for all {{math|1 ≤ ''k'' < ''n''}}.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Example 6 in §7.3}} for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.</ref> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A [[spanning tree]] is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees {{math|''fn''}} of a fan of order {{mvar|n}} are possible for each {{math|''n'' ≥ 1}}.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when {{math|''n'' {{=}} 4}}, we have that {{math|''f''4 {{=}} 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 {{=}} 21}}, which is a sum over the {{mvar|m}}-fold convolutions of the sequence {{math|''gn'' {{=}} ''n'' {{=}} [''zn''] {{sfrac|''z''|(1 − ''z'')2}}}} for {{math|''m'' ≔ 1, 2, 3, 4}}. More generally, we may write a formula for this sequence as
<math display="block">f_n = \sum_{m > 0} \sum_{\scriptstyle k_1+k_2+\cdots+k_m=n\atop\scriptstyle k_1, k_2, \ldots,k_m > 0} g_{k_1} g_{k_2} \cdots g_{k_m}\,, </math>
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
<math display="block">F(z) = G(z) + G(z)^2 + G(z)^3 + \cdots = \frac{G(z)}{1-G(z)} = \frac{z}{(1-z)^2-z} = \frac{z}{1-3z+z^2}\,,</math>
from which we are able to extract an exact formula for the sequence by taking the [[partial fraction expansion]] of the last generating function.

===Implicit generating functions and the Lagrange inversion formula===
{{expand section|This section needs to be added to the list of techniques with generating functions|date=April 2017}}

===Introducing a free parameter (snake oil method)===
Sometimes the sum {{math|''sn''}} is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have {{mvar|n}} as limit in the summation. When n does not appear explicitly in the summation, we may consider {{mvar|n}} as a "free" parameter and treat {{math|''sn''}} as a coefficient of {{math|''F''(''z'') {{=}} Σ ''sn'' ''zn''}}, change the order of the summations on {{mvar|n}} and {{mvar|k}}, and try to compute the inner sum.

For example, if we want to compute
<math display="block">s_n = \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\,, \quad m,n \in \mathbb{N}_0\,,</math>
we can treat {{mvar|n}} as a "free" parameter, and set
<math display="block">F(z) = \sum_{n = 0}^\infty{\left( \sum_{k = 0}^\infty{\binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1}}\right) }z^n\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">F(z) = \sum_{k = 0}^\infty{\binom{2k}{k}\frac{(-1)^k}{k+1} z^{-k}}\sum_{n = 0}^\infty{\binom{n+k}{m+2k} z^{n+k}}\,.</math>

Now the inner sum is {{math|{{sfrac|''z''''m'' + 2''k''|(1 − ''z'')''m'' + 2''k'' + 1}}}}. Thus
<math display="block">\begin{align} F(z)
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{\frac{1}{k+1}\binom{2k}{k}\left(\frac{-z}{(1-z)^2}\right)^k} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\sum_{k = 0}^\infty{C_k\left(\frac{-z}{(1-z)^2}\right)^k} &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= \frac{z^m}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^2}}}{\frac{-2z}{(1-z)^2}} \\[4px]
&= \frac{-z^{m-1}}{2(1-z)^{m-1}}\left(1-\frac{1+z}{1-z}\right) \\[4px]
&= \frac{z^m}{(1-z)^m} = z\frac{z^{m-1}}{(1-z)^m}\,.
\end{align}</math>

Then we obtain
<math display="block">s_n = \begin{cases}
\displaystyle\binom{n-1}{m-1} & \text{for } m \geq 1 \,, \\ {}
[n = 0] & \text{for } m = 0\,.
\end{cases}</math>

It is instructive to use the same method again for the sum, but this time take {{mvar|m}} as the free parameter instead of {{mvar|n}}. We thus set
<math display="block">G(z) = \sum_{m = 0}^\infty\left( \sum_{k = 0}^\infty \binom{n+k}{m+2k}\binom{2k}{k}\frac{(-1)^k}{k+1} \right) z^m\,.</math>

Interchanging summation ("snake oil") gives
<math display="block">G(z) = \sum_{k = 0}^\infty \binom{2k}{k}\frac{(-1)^k}{k+1} z^{-2k} \sum_{m = 0}^\infty \binom{n+k}{m+2k} z^{m+2k}\,.</math>

Now the inner sum is {{math|(1 + ''z'')''n'' + ''k''}}. Thus
<math display="block">\begin{align} G(z)
&= (1+z)^n \sum_{k = 0}^\infty \frac{1}{k+1}\binom{2k}{k}\left(\frac{-(1+z)}{z^2}\right)^k \\[4px]
&= (1+z)^n \sum_{k = 0}^\infty C_k \,\left(\frac{-(1+z)}{z^2}\right)^k &\text{where } C_k = k\text{th Catalan number} \\[4px]
&= (1+z)^n \,\frac{1-\sqrt{1+\frac{4(1+z)}{z^2}}}{\frac{-2(1+z)}{z^2}} \\[4px]
&= (1+z)^n \,\frac{z^2-z\sqrt{z^2+4+4z}}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{z^2-z(z+2)}{-2(1+z)} \\[4px]
&= (1+z)^n \,\frac{-2z}{-2(1+z)} = z(1+z)^{n-1}\,.
\end{align}</math>

Thus we obtain
<math display="block">s_n = \left[z^m\right] z(1+z)^{n-1} = \left[z^{m-1}\right] (1+z)^{n-1} = \binom{n-1}{m-1}\,,</math>
for {{math|''m'' ≥ 1}} as before.

===Generating functions prove congruences===
We say that two generating functions (power series) are congruent modulo {{mvar|m}}, written {{math|''A''(''z'') ≡ ''B''(''z'') (mod ''m'')}} if their coefficients are congruent modulo {{mvar|m}} for all {{math|''n'' ≥ 0}}, i.e., {{math|''an'' ≡ ''bn'' (mod ''m'')}} for all relevant cases of the integers {{mvar|n}} (note that we need not assume that {{mvar|m}} is an integer here—it may very well be polynomial-valued in some indeterminate {{mvar|x}}, for example). If the "simpler" right-hand-side generating function, {{math|''B''(''z'')}}, is a rational function of {{mvar|z}}, then the form of this sequence suggests that the sequence is [[periodic function|eventually periodic]] modulo fixed particular cases of integer-valued {{math|''m'' ≥ 2}}. For example, we can prove that the [[Euler numbers]],
<math display="block">\langle E_n \rangle = \langle 1, 1, 5, 61, 1385, \ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle \pmod{3}\,,</math>
satisfy the following congruence modulo 3:<ref>{{harvnb|Lando|2003|loc=§5}}</ref>
<math display="block">\sum_{n = 0}^\infty E_n z^n = \frac{1-z^2}{1+z^2} \pmod{3}\,. </math>

One of the most useful, if not downright powerful, methods of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers {{math|''pk''}}) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by {{mvar|J}}-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's ''Lectures on Generating Functions'' as follows:
{{math theorem | name = Theorem: congruences for series generated by expansions of continued fractions
| math_statement = Suppose that the generating function {{math|''A''(''z'')}} is represented by an infinite [[continued fraction]] of the form
<math display="block">A(z) = \cfrac{1}{1-c_1z - \cfrac{p_1z^2}{1-c_2z - \cfrac{p_2 z^2}{1-c_3z - {\ddots}}}}</math>
and that {{math|''Ap''(''z'')}} denotes the {{mvar|p}}th convergent to this continued fraction expansion defined such that {{math|''an'' {{=}} [''zn''] ''Ap''(''z'')}} for all {{math|0 ≤ ''n'' < 2''p''}}. Then:

# the function {{math|''Ap''(''z'')}} is rational for all {{math|''p'' ≥ 2}} where we assume that one of divisibility criteria of {{math|''p'' {{!}} ''p''1, ''p''1''p''2, ''p''1''p''2''p''3}} is met, that is, {{math|''p'' {{!}} ''p''1''p''2⋯''p''''k''}} for some {{math|''k'' ≥ 1}}; and
# if the integer {{mvar|p}} divides the product {{math|''p''1''p''2⋯''p''''k''}}, then we have {{math|''A''(''z'') ≡ ''Ak''(''z'') (mod ''p'')}}.}}

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the [[Stirling numbers of the first kind]] and for the [[partition function (mathematics)|partition function {{math|''p''(''n'')}}]] which show the versatility of generating functions in tackling problems involving [[integer sequences]].

====The Stirling numbers modulo small integers====

The [[Stirling numbers of the first kind#Congruences|main article]] on the Stirling numbers generated by the finite products
<math display="block">S_n(x) := \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k = x(x+1)(x+2) \cdots (x+n-1)\,,\quad n \geq 1\,, </math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference ''Generatingfunctionology''.
We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<math display="block">S_n(x) = [x(x+1)] \cdot [x(x+1)] \cdots = x^{\left\lceil \frac{n}{2} \right\rceil} (x+1)^{\left\lfloor \frac{n}{2} \right\rfloor}\,, </math>

which implies that the parity of these [[Stirling numbers]] matches that of the binomial coefficient

<math display="block">\begin{bmatrix} n \\ k \end{bmatrix} \equiv \binom{\left\lfloor \frac{n}{2} \right\rfloor}{k - \left\lceil \frac{n}{2} \right\rceil} \pmod{2}\,, </math>

and consequently shows that {{math|{{resize|150%|[}}{{su|p=''n''|b=''k''|a=c}}{{resize|150%|]}}}} is even whenever {{math|''k'' < ⌊ {{sfrac|''n''|2}} ⌋}}.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that
<math display="block">\begin{align}
\begin{bmatrix} n \\ m \end{bmatrix} & \equiv
[x^m] \left(
x^{\left\lceil \frac{n}{3} \right\rceil} (x+1)^{\left\lceil \frac{n-1}{3} \right\rceil}
(x+2)^{\left\lfloor \frac{n}{3} \right\rfloor}
\right) && \pmod{3} \\
& \equiv
\sum_{k=0}^{m} \begin{pmatrix} \left\lceil \frac{n-1}{3} \right\rceil \\ k \end{pmatrix}
\begin{pmatrix} \left\lfloor \frac{n}{3} \right\rfloor \\ m-k - \left\lceil \frac{n}{3} \right\rceil \end{pmatrix} \times
2^{\left\lceil \frac{n}{3} \right\rceil + \left\lfloor \frac{n}{3} \right\rfloor -(m-k)} && \pmod{3}\,.
\end{align}</math>

====Congruences for the partition function====

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that ''the'' [[partition function (number theory)|partition function]] {{math|''p''(''n'')}} is generated by the reciprocal infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product (or {{mvar|z}}-Pochhammer product as the case may be) given by
<math display="block">\begin{align}
\sum_{n = 0}^\infty p(n) z^n & = \frac{1}{\left(1-z\right)\left(1-z^2\right)\left(1-z^3\right) \cdots} \\[4pt]
& = 1 + z + 2z^2 + 3 z^3 + 5z^4 + 7z^5 + 11z^6 + \cdots.
\end{align}</math>

This partition function satisfies many known [[Ramanujan's congruences|congruence properties]], which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<ref>{{harvnb|Hardy|Wright|Heath-Brown|Silverman|2008|loc=§19.12}}</ref>
<math display="block">\begin{align}
p(5m+4) & \equiv 0 \pmod{5} \\
p(7m+5) & \equiv 0 \pmod{7} \\
p(11m+6) & \equiv 0 \pmod{11} \\
p(25m+24) & \equiv 0 \pmod{5^2}\,.
\end{align}</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function
<math display=block>\frac{1}{(1-z)^5} = \sum_{i=0}^\infty \binom{4+i}{4}z^i\,,</math>
all of the coefficients are divisible by 5 except for those which correspond to the powers {{math|1, ''z''5, ''z''10, ...}} and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
<math display="block">\frac{1}{(1-z)^5} \equiv \frac{1}{1-z^5} \pmod{5}\,,</math>
or equivalently
<math display="block"> \frac{1-z^5}{(1-z)^5} \equiv 1 \pmod{5}\,.</math>
It follows that
<math display="block">\frac{\left(1-z^5\right)\left(1-z^{10}\right)\left(1-z^{15}\right) \cdots }{\left((1-z)\left(1-z^2\right)\left(1-z^3\right) \cdots \right)^5} \equiv 1 \pmod{5}\,. </math>

Using the infinite product expansions of
<math display="block">z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(1-z\right)\left(1-z^2\right) \cdots } =
z \cdot \left((1-z)\left(1-z^2\right) \cdots \right)^4 \times \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{\left(\left(1-z\right)\left(1-z^2\right) \cdots \right)^5}\,,</math>
it can be shown that the coefficient of {{math|''z''5''m'' + 5}} in {{math|''z'' · ((1 − ''z'')(1 − ''z''2)⋯)4}} is divisible by 5 for all {{mvar|m}}.<ref>{{cite book |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M.|title=An Introduction to the Theory of Numbers}} p.288, Th.361</ref> Finally, since
<math display="block">\begin{align}
\sum_{n = 1}^\infty p(n-1) z^n & = \frac{z}{(1-z)\left(1-z^2\right) \cdots} \\[6px]
& = z \cdot \frac{\left(1-z^5\right)\left(1-z^{10}\right) \cdots }{(1-z)\left(1-z^2\right) \cdots } \times \left(1+z^5+z^{10}+\cdots\right)\left(1+z^{10}+z^{20}+\cdots\right) \cdots
\end{align}</math>
we may equate the coefficients of {{math|''z''5''m'' + 5}} in the previous equations to prove our desired congruence result, namely that {{math|''p''(5''m'' + 4) ≡ 0 (mod 5)}} for all {{math|''m'' ≥ 0}}.

===Transformations of generating functions===
There are a number of transformations of generating functions that provide other applications (see the [[generating function transformation|main article]]). A transformation of a sequence's ''ordinary generating function'' (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see [[Generating function transformation#Integral Transformations|integral transformations]]) or weighted sums over the higher-order derivatives of these functions (see [[Generating function transformation#Derivative Transformations|derivative transformations]]).

Generating function transformations can come into play when we seek to express a generating function for the sums

<math display="block">s_n := \sum_{m=0}^n \binom{n}{m} C_{n,m} a_m, </math>

in the form of {{math|''S''(''z'') {{=}} ''g''(''z'') ''A''(''f''(''z''))}} involving the original sequence generating function. For example, if the sums are
<math display="block">s_n := \sum_{k = 0}^\infty \binom{n+k}{m+2k} a_k \,</math>
then the generating function for the modified sum expressions is given by<ref>{{harvnb|Graham|Knuth|Patashnik|1994|p=535, exercise 5.71}}</ref>
<math display="block">S(z) = \frac{z^m}{(1-z)^{m+1}} A\left(\frac{z}{(1-z)^2}\right)</math>
(see also the [[binomial transform]] and the [[Stirling transform]]).

There are also integral formulas for converting between a sequence's OGF, {{math|''F''(''z'')}}, and its exponential generating function, or EGF, {{math|''F̂''(''z'')}}, and vice versa given by

<math display="block">\begin{align}
F(z) &= \int_0^\infty \hat{F}(tz) e^{-t} \, dt \,, \\[4px]
\hat{F}(z) &= \frac{1}{2\pi} \int_{-\pi}^\pi F\left(z e^{-i\vartheta}\right) e^{e^{i\vartheta}} \, d\vartheta \,,
\end{align}</math>

provided that these integrals converge for appropriate values of {{mvar|z}}.

===Other applications===
Generating functions are used to:

* Find a [[closed formula]] for a sequence given in a recurrence relation. For example, consider [[Fibonacci number#Generating function|Fibonacci numbers]].
* Find [[recurrence relation]]s for sequences—the form of a generating function may suggest a recurrence formula.
* Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.
* Explore the asymptotic behaviour of sequences.
* Prove identities involving sequences.
* Solve [[enumeration]] problems in [[combinatorics]] and encoding their solutions. [[Rook polynomial]]s are an example of an application in combinatorics.
* Evaluate infinite sums.

==Other generating functions==

===Examples===

Examples of [[polynomial sequence]]s generated by more complex generating functions include:

* [[Appell polynomials]]
* [[Chebyshev polynomials]]
* [[Difference polynomials]]
* [[Generalized Appell polynomials]]
* [[Q-difference polynomial|{{mvar|q}}-difference polynomials]]

Other sequences generated by more complex generating functions:

* Double exponential generating functions. For example: [https://oeis.org/search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search Aitken's Array: Triangle of Numbers]
* Hadamard products of generating functions and diagonal generating functions, and their corresponding [[generating function transformation#Hadamard products and diagonal generating functions|integral transformations]]

===Convolution polynomials===

Knuth's article titled "''Convolution Polynomials''"<ref>{{cite journal|last1=Knuth|first1=D. E.|title=Convolution Polynomials|journal=Mathematica J.|date=1992|volume=2|pages=67–78|arxiv=math/9207221|bibcode=1992math......7221K}}</ref> defines a generalized class of ''convolution polynomial'' sequences by their special generating functions of the form
<math display="block">F(z)^x = \exp\bigl(x \log F(z)\bigr) = \sum_{n = 0}^\infty f_n(x) z^n,</math>
for some analytic function {{mvar|F}} with a power series expansion such that {{math|''F''(0) {{=}} 1}}.

We say that a family of polynomials, {{math|''f''0, ''f''1, ''f''2, ...}}, forms a ''convolution family'' if {{math|[[Degree of a polynomial|deg]] ''fn'' ≤ ''n''}} and if the following convolution condition holds for all {{mvar|x}}, {{mvar|y}} and for all {{math|''n'' ≥ 0}}:
<math display="block">f_n(x+y) = f_n(x) f_0(y) + f_{n-1}(x) f_1(y) + \cdots + f_1(x) f_{n-1}(y) + f_0(x) f_n(y). </math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

* The sequence {{math|''n''! · ''fn''(''x'')}} is of [[binomial type]]
* Special values of the sequence include {{math|''fn''(1) {{=}} [''zn''] ''F''(''z'')}} and {{math|''fn''(0) {{=}} ''δ''''n'',0}}, and
* For arbitrary (fixed) <math>x, y, t \isin \mathbb{C}</math>, these polynomials satisfy convolution formulas of the form
<math display="block">\begin{align}
f_n(x+y) & = \sum_{k=0}^n f_k(x) f_{n-k}(y) \\
f_n(2x) & = \sum_{k=0}^n f_k(x) f_{n-k}(x) \\
xn f_n(x+y) & = (x+y) \sum_{k=0}^n k f_k(x) f_{n-k}(y) \\
\frac{(x+y) f_n(x+y+tn)}{x+y+tn} & = \sum_{k=0}^n \frac{x f_k(x+tk)}{x+tk} \frac{y f_{n-k}(y+t(n-k))}{y+t(n-k)}.
\end{align}</math>

For a fixed non-zero parameter <math>t \isin \mathbb{C}</math>, we have modified generating functions for these convolution polynomial sequences given by
<math display="block">\frac{z F_n(x+tn)}{(x+tn)} = \left[z^n\right] \mathcal{F}_t(z)^x, </math>
where {{math|𝓕''t''(''z'')}} is implicitly defined by a [[functional equation]] of the form {{math|𝓕''t''(''z'') {{=}} ''F''(''x''𝓕''t''(''z'')''t'')}}. Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, {{math|⟨ ''fn''(''x'') ⟩}} and {{math|⟨ ''gn''(''x'') ⟩}}, with respective corresponding generating functions, {{math|''F''(''z'')''x''}} and {{math|''G''(''z'')''x''}}, then for arbitrary {{mvar|t}} we have the identity
<math display="block">\left[z^n\right] \left(G(z) F\left(z G(z)^t\right)\right)^x = \sum_{k=0}^n F_k(x) G_{n-k}(x+tk). </math>

Examples of convolution polynomial sequences include the ''binomial power series'', {{math|𝓑''t''(''z'') {{=}} 1 + ''z''𝓑''t''(''z'')''t''}}, so-termed ''tree polynomials'', the [[Bell numbers]], {{math|''B''(''n'')}}, the [[Laguerre polynomials]], and the [[Stirling polynomial|Stirling convolution polynomials]].

===Tables of special generating functions===

An initial listing of special mathematical series is found [[List of mathematical series|here]]. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of ''Concrete Mathematics'' and in Section 2.5 of Wilf's ''Generatingfunctionology''. Other special generating functions of note include the entries in the next table, which is by no means complete.<ref>See also the ''1031 Generating Functions'' found in {{cite thesis |first=Simon |last=Plouffe |title=Approximations de séries génératrices et quelques conjectures |trans-title=Approximations of generating functions and a few conjectures |year=1992 |type=Masters |publisher=Université du Québec à Montréal |language=fr |arxiv=0911.4975}}</ref>

{{expand section|Lists of special and special sequence generating functions. The next table is a start|date=April 2017}}

:{| class="wikitable"
|-
! Formal power series !! Generating-function formula !! Notes
|-
| <math>\sum_{n = 0}^\infty \binom{m+n}{n} \left(H_{n+m}-H_m\right) z^n</math> || <math>\frac{1}{(1-z)^{m+1}} \ln \frac{1}{1-z}</math> || <math>H_n</math> is a first-order [[harmonic number]]
|-
| <math>\sum_{n = 0}^\infty B_n \frac{z^n}{n!}</math> || <math>\frac{z}{e^z-1}</math> || <math>B_n</math> is a [[Bernoulli number]]
|-
| <math>\sum_{n = 0}^\infty F_{mn} z^n</math> || <math>\frac{F_m z}{1-(F_{m-1}+F_{m+1})z+(-1)^m z^2}</math> || <math>F_n</math> is a [[Fibonacci number]] and <math>m \in \mathbb{Z}^{+}</math>
|-
| <math>\sum_{n = 0}^\infty \left\{\begin{matrix} n \\ m \end{matrix} \right\} z^n</math> || <math>(z^{-1})^{\overline{-m}} = \frac{z^m}{(1-z)(1-2z)\cdots(1-mz)}</math> || <math>x^{\overline{n}}</math> denotes the [[rising factorial]], or [[Pochhammer symbol]] and some integer <math>m \geq 0</math>
|-
| <math>\sum_{n = 0}^\infty \left[\begin{matrix} n \\ m \end{matrix} \right] z^n</math> || <math>z^{\overline{m}} = z(z+1) \cdots (z+m-1)</math>
|-
| <math>\sum_{n = 1}^\infty \frac{(-1)^{n-1}4^n (4^n-2) B_{2n} z^{2n}}{(2n) \cdot (2n)!}</math> || <math>\ln \frac{\tan(z)}{z}</math>
|-
| <math>\sum_{n = 0}^\infty \frac{(1/2)^{\overline{n}} z^{2n}}{(2n+1) \cdot n!}</math> || <math>z^{-1} \arcsin(z)</math>
|-
| <math>\sum_{n = 0}^\infty H_n^{(s)} z^n</math> || <math>\frac{\operatorname{Li}_s(z)}{1-z}</math> || <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]] function and <math>H_n^{(s)}</math> is a generalized [[harmonic number]] for <math>\Re(s) > 1</math>
|-
| <math>\sum_{n = 0}^\infty n^m z^n</math> || <math>\sum_{0 \leq j \leq m} \left\{\begin{matrix} m \\ j \end{matrix} \right\} \frac{j! \cdot z^j}{(1-z)^{j+1}}</math> || <math>\left\{\begin{matrix} n \\ m \end{matrix} \right\}</math> is a [[Stirling number of the second kind]] and where the individual terms in the expansion satisfy <math>\frac{z^i}{(1-z)^{i+1}} = \sum_{k=0}^{i} \binom{i}{k} \frac{(-1)^{k-i}}{(1-z)^{k+1}}</math>
|-
| <math>\sum_{k < n} \binom{n-k}{k} \frac{n}{n-k} z^k</math> || <math>\left(\frac{1+\sqrt{1+4z}}{2}\right)^n + \left(\frac{1-\sqrt{1+4z}}{2}\right)^n</math> ||
|-
| <math>\sum_{n_1, \ldots, n_m \geq 0} \min(n_1, \ldots, n_m) z_1^{n_1} \cdots z_m^{n_m}</math> || <math>\frac{z_1 \cdots z_m}{(1-z_1) \cdots (1-z_m) (1-z_1 \cdots z_m)}</math> || The two-variable case is given by <math>M(w, z) := \sum_{m,n \geq 0} \min(m, n) w^m z^n = \frac{wz}{(1-w)(1-z)(1-wz)}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{s}{n} z^n</math> || <math>(1+z)^s</math> || <math>s \in \mathbb{C}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{n}{k} z^n</math> || <math>\frac{z^k}{(1-z)^{k+1}}</math> || <math>k \in \mathbb{N}</math>
|-
|<math>\sum_{n = 1}^\infty \log{(n)} z^n</math>||<math>\left.-\frac{\partial}{\partial s}\operatorname{{Li}_s(z)}\right|_{s=0}</math>||
|}

== History ==
[[George Pólya]] writes in ''[[Mathematics and plausible reasoning]]'':
<blockquote>''The name "generating function" is due to [[Laplace]]. Yet, without giving it a name, [[Euler]] used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the [[Number theory|Theory of Numbers]].''</blockquote>

==See also==
* [[Moment-generating function]]
* [[Probability-generating function]]
* [[Generating function transformation]]
* [[Stanley's reciprocity theorem]]
* Applications to [[Partition (number theory)]]
* [[Combinatorial principles]]
* [[Cyclic sieving]]
* [[Z-transform]]
* [[Umbral calculus]]

==Notes==
{{noteFoot}}

==References==
{{reflist}}

===Citations===
*{{cite book |first=Martin |last=Aigner |title=A Course in Enumeration |url=https://books.google.com/books?id=pPEJcu93dzAC |date=2007 |publisher=Springer |isbn=978-3-540-39035-0 |series=Graduate Texts in Mathematics |volume=238 }}
* {{cite journal |title=On the foundations of combinatorial theory. VI. The idea of generating function |last1=Doubilet |first1=Peter |last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | last3=Stanley | first3=Richard | author3-link=Richard P. Stanley | journal=Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability |volume=2 |pages=267–318 |year=1972 | zbl=0267.05002 | url=http://projecteuclid.org/euclid.bsmsp/1200514223 }} Reprinted in {{cite book | last=Rota | first=Gian-Carlo | author-link=Gian-Carlo Rota | others=With the collaboration of P. Doubilet, C. Greene, D. Kahaner, [[Andrew Odlyzko|A. Odlyzko]] and [[Richard P. Stanley|R. Stanley]] | title=Finite Operator Calculus | chapter=3. The idea of generating function | pages=83–134 | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 | zbl=0328.05007 }}
* {{cite book | last1 = Flajolet | first1 = Philippe | author-link1 = Philippe Flajolet | last2 = Sedgewick | first2 = Robert | author-link2 = Robert Sedgewick (computer scientist) | title = Analytic Combinatorics | title-link= Analytic Combinatorics | year = 2009 | publisher = Cambridge University Press | isbn = 978-0-521-89806-5 | zbl=1165.05001 }}
* {{cite book | last1 = Goulden | first1 = Ian P. | last2 = Jackson | first2 = David M. | author-link2 = David M. Jackson | title = Combinatorial Enumeration | year = 2004 | publisher = [[Dover Publications]] | isbn = 978-0486435978 }}
* {{cite book |title=[[Concrete Mathematics|Concrete Mathematics. A foundation for computer science]] |edition=2nd |year=1994 |publisher=Addison-Wesley |isbn=0-201-55802-5 |chapter=Chapter 7: Generating Functions |pages=320–380| zbl=0836.00001 |first1 = Ronald L. |last1=Graham |first2 = Donald E. |last2=Knuth |first3=Oren |last3=Patashnik |author-link1=Ronald Graham |author-link2=Donald Knuth |author-link3=Oren Patashnik }}
*{{cite book |first=Sergei K. |last=Lando |title=Lectures on Generating Functions |url=https://books.google.com/books?id=A6_4AwAAQBAJ |date=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3481-7 }}
* {{cite book | last=Wilf | first=Herbert S. | author-link=Herbert Wilf | title=Generatingfunctionology | edition=2nd | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 | url=http://www.math.upenn.edu/%7Ewilf/DownldGF.html }}

==External links==
* [http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf "Introduction To Ordinary Generating Functions"] by Mike Zabrocki, York University, Mathematics and Statistics
* {{springer|title=Generating function|id=p/g043900}}
* [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/GeneratingFunctions/ "Generating Functions"] by [[Ed Pegg Jr.]], [[Wolfram Demonstrations Project]], 2007.

{{Authority control}}

{{DEFAULTSORT:Generating Function}}
[[Category:1730 introductions]]
[[Category:Generating functions| ]]
[[Category:Abraham de Moivre]]

Talk:Exact sequence

2023-11-24T05:16:59Z

Yeetcode: /* Properties */ new section

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== 2Z --> Z or Z -->2Z ==
there were a couple of corrections by anonymous editors recently that I've just reverted. There seem to be three different choices for
the example exact sequence:

# 0 → '''Z''' → '''Z''' → '''Z'''/2'''Z''' → 0
# 0 → '''Z''' → 2'''Z''' → '''Z'''/2'''Z''' → 0
# 0 → 2'''Z''' → '''Z''' → '''Z'''/2'''Z''' → 0

The first two are pretty much the same, the second map is ''n'' to 2''n'', and the only question is how you want to label it. The third one is slightly different, the second arrow is an inclusion map. The anonymous editors have gone through all three, and I reverted back to the original, which is #1. But actually, I prefer #3, because it shows more explicitly the general paradigm that for any quotient group ''B''/''A'', you have an exact sequence 1 → ''A'' → ''B'' → ''B''/''A'' → 1, whereas the other sequences don't have the names in the right places. I wonder what others think. -[[User:Lethe/sig|lethe]] [[User talk:Lethe/sig|talk]] 01:26, 27 January 2006 (UTC)
:Actually, I think the second one is wrong: the image of 2'''Z''' → '''Z'''/2'''Z''' is 0, while the kernel of '''Z'''/2'''Z''' → 0 is {0,1}, so that's not exact. -[[User:Lethe/sig|lethe]] [[User talk:Lethe/sig|talk]] 07:05, 27 January 2006 (UTC)
::The second one could be correct, but the map <math> \mathbb{Z} \to 2 \mathbb{Z}</math> would have to be <math>n \mapsto 4n</math> (or n goes to -4n), and that seems kind of pointless. [[Special:Contributions/156.56.139.205|156.56.139.205]] ([[User talk:156.56.139.205|talk]]) 14:44, 13 September 2011 (UTC)

I prefer the first because it keeps the external diagram external. 2'''Z''' makes sense as the kernel in the quotient '''Z'''/2'''Z''', but is uneccessary if not confusing as the second group in #3. [[User:MotherFunctor|MotherFunctor]] 06:01, 14 May 2006 (UTC)
:I'm not sure what you mean by "external diagram". Can you explain? Cute handle by the way. -[[User:Lethe/sig|lethe]] [[User talk:Lethe/sig|talk]] [{{fullurl:User talk:Lethe|action=edit&section=new}} +] 06:42, 14 May 2006 (UTC)

:: Thanks and sure. It comes from a nice categorical set theory book "Sets For Mathematics" Lawvere, Rosenbrugh. External diagram labels objects and arrows, internal diagram shows behavior of arrows on points in object. <math>\mathbb Z</math> and <math>\mathbb Z/\mathbb Z_2 </math> are objects. <math>n\mathbb Z</math> is not an object, unless it's another name for <math>\mathbb Z</math>. Anyway, I think it is bad style, as is evident from the confusion. The first one is nice. [[User:MotherFunctor|MotherFunctor]] 05:46, 17 May 2006 (UTC)

The version currently in the article is much the best:

# 0 → '''Z''' → '''Z''' → '''Z'''/2'''Z''' → 0

The problem with the other two is that they try to make the names of objects stand in for the names of functions. There is no doubling involved in either of the two copies of '''Z''' but rather inthe function between them.[[User:Colin McLarty|Colin McLarty]] ([[User talk:Colin McLarty|talk]]) 22:46, 2 June 2010 (UTC)

This seems to be one of those holy topics that Wikipedians forever argue about. I think you're more likely to see the first half as

:<math>2\mathbb{Z} \;{\hookrightarrow}\; \mathbb{Z} \twoheadrightarrow \mathbb{Z}/2\mathbb{Z}</math>

in most math books with the inclusion being simply the (deceivingly "identity"-like) map <math>2n \mapsto 2n</math>. The problem with the current presentation is that's not clear how the Z ends up being 2Z until you specify the function, while with this version the function should be said in text for completeness, but it's mostly obvious. [[Special:Contributions/86.127.138.67|86.127.138.67]] ([[User talk:86.127.138.67|talk]]) 19:50, 4 April 2015 (UTC)

== equalizers ==
As pointed out on the talk page to [[sheaf (mathematics)]], it is often the case that one of the arrows is an [[equalizer]], i.e. there are also two parallel arrows, and that this is how the [[Mayer-Vietoris sequence]] is constructed. It would be nice if some kind of explicit discussion of this case was handled here. [[User:Linas|linas]] ([[User talk:Linas|talk]]) 21:57, 18 August 2012 (UTC)
:After some digging, it appears that the [[coequalizer]] article provides the needed statement that its a generalization of the idea of a quotient. Then, in the examples section, it even gives a the standard homological example of gluing two arcs together to make S^1. Yay! What we need now is to transpose all of that into this article... [[User:Linas|linas]] ([[User talk:Linas|talk]]) 23:02, 18 August 2012 (UTC)

== Short exact sequences ==
This article is OK, especially the examples grad ⇒ rot ⇒ div are really nice. But it should be augmented by the fact, that short exact sequences are equivalently defined by a pair of functions with some properties, which in the split case often is used for the definition of semi-direct products (or sums). So the relation to the 2nd cohomology can be given more explicitely. In addition if the definition of split is applied to the other morphism in the sh.ex.seq then semi-direct reduces to direct. This case some-times is called "retract". — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/134.60.206.14|134.60.206.14]] ([[User talk:134.60.206.14|talk]]) 11:05, 11 April 2013 (UTC) 

== {{xtag|ce}} markup for automatic tuning of arrow lengths and spaces ==
I found the way that {{xtag|ce}} markup can tune arrow lengths and spaces automatically. -- [[User:Cedar101|Cedar101]] ([[User talk:Cedar101|talk]]) 09:56, 25 January 2018 (UTC)
{{markup|{{code|2=tex|0 -> \mathit{A ->[~~f~~] B ->[~~g~~] C} -> 0}}
|<ce>0 -> \mathit{A ->[~~f~~] B ->[~~g~~] C} -> 0</ce>
|{{code|2=tex|\mathbb{H1 ->[grad] H_{curl} ->[curl] H_{div} ->[div] L2} }}
|<ce>\mathbb{H1 ->[grad] H_{curl} ->[curl] H_{div} ->[div] L2}</ce>}}
:Wow! Thanks! Cool! I always wondered how to do that! Different question ... What's H_1 and L_2 and what's Hilbert spaces got to do with it? (I assume you added the above content to the article, which mentions Hilbert spaces...) [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 06:59, 9 May 2019 (UTC)

== Link ==

Someone reverted the addition of:

--External links--
[https://www.youtube.com/watch?v=rXEiJhBHJsU Short Exact Sequences], explanation by Matthew Salomone

I think that's a shame because it gives a much better explanation than anything contained in the article, which is not very well written. Perhaps it should be restored?

[[User:Stikko|Stikko]] ([[User talk:Stikko|talk]]) 21:44, 26 September 2021 (UTC)
:Please, read [[WP:EL]], and specifically the first item of [[WP:ELNO]] ({{tqq|One should generally avoid providing external links to [...] any site that does not provide a unique resource beyond what the article would contain if it became a featured article. In other words, the site should not merely repeat information that is already or should be in the article}}). This applies to this external link. Also [[WP:NOR]] applies to this video, which, in any case is not a [[WP:reliable sources|reliable source for Wikipedia]]. More specifically, in mathematics, YouTube videos are generally not accepted, except in very exceptional cases. Instead of trying to link this YouTube video, I suggest you to use it for proposing here specific improvements to the article. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:20, 27 September 2021 (UTC)

== Properties ==

At the beginning of the section, there is the claim that "for non-commutative groups, this is the semidirect product". It seems straight up incorrect. I do not think semidirect products are any kind of products for starters: One does not have uniqueness without a pre-specified homomorphism
<math>\phi : H \to Aut(K)</math>. I could still be missing something, but the amount of clarification is rather inadequate. [[User:Yeetcode|Yeetcode]] ([[User talk:Yeetcode|talk]]) 05:16, 24 November 2023 (UTC)