Chebyshev polynomials

The Chebyshev polynomials are named after Pafnuty Chebyshev (Пафнутий Чебышёв) and compose a polynomial sequence. This article refers to what are commonly known as Chebyshev polynomials of the first kind, which are a solution to the Chebyshev differential equation:

$(1-x^{2})\,y''-x\,y'+n^{2}\,y=0$

They can be defined by

$T_{n}(\cos(\theta ))=\cos(n\theta )\,$

for n = 0, 1, 2, 3, .... . That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of De Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos²(x) + sin&sup2(x) = 1. The polynomial T_n has exactly n simple roots in [−1, 1] called Chebyshev roots. The Chebyshev polynomials can be used in the area of numerical approximation.

Alternatively they can be defined via the recurrence relation

$T_{0}(x)=1\,$

$T_{1}(x)=x\,$

$T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x)\,$

These polynomials are orthogonal with respect to the weight

${\frac {dx}{\sqrt {1-x^{2}}}},$

on the interval [−1,1], i.e., we have

$\int _{-1}^{1}T_{n}(x)T_{m}(x)\,{\frac {dx}{\sqrt {1-x^{2}}}}=0\quad {\mbox{if}}\ n\neq m.$

This is because (letting x = cos θ)

$\int _{0}^{\pi }\cos(n\theta )\cos(m\theta )\,d\theta =0\quad {\mbox{if}}\ n\neq m.$

The first few polynomials are:

$T_{0}(x)=1\,$

$T_{1}(x)=x\,$

$T_{2}(x)=2x^{2}-1\,$

$T_{3}(x)=4x^{3}-3x\,$

$T_{4}(x)=8x^{4}-8x^{2}+1\,$

$T_{5}(x)=16x^{5}-20x^{3}+5x\,$

$T_{6}(x)=32x^{6}-48x^{4}+18x^{2}-1\,$

$T_{7}(x)=64x^{7}-112x^{5}+56x^{3}-7x\,$

$T_{8}(x)=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\,$

$T_{9}(x)=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\,$

One example of a generating function for this polynomial sequence is

$\sum _{n=0}^{\infty }T_{n}(x)t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.$

See also

Chebyshev nodes

Legendre polynomials

Hermite polynomials