Definitions

# Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. They are named for Leopold Gegenbauer (1849-1903). They are obtained from hypergeometric series in cases where the series is in fact finite:

$C_n^\left\{\left(alpha\right)\right\}\left(z\right)=frac\left\{\left(2alpha\right)^\left\{underline\left\{n\right\}\right\}\right\}\left\{n!\right\}$
,_2F_1left(-n,2alpha+n;alpha+frac{1}{2};frac{1-z}{2}right)

where $underline\left\{n\right\}$ is the falling factorial. (Abramowitz & Stegun p561)

Gegenbauer polynomials appear from solving the Gegenbauer differential equation:

$\left(1-x^\left\{2\right\}\right)y\text{'}\text{'}-\left(2n+3\right)xy\text{'}+\left\{alpha\right\}y=0$

They are closely related to ultraspherical polynomials and can be viewed as an extension of the Legendre polynomials, since they can be obtained from the generating function:

$frac\left\{1\right\}\left\{\left(1-2xt+t^\left\{2\right\}\right)^\left\{alpha\right\}\right\}=sum_\left\{n=0\right\}^\left\{infty\right\}C_n^\left\{\left(alpha\right)\right\}\left(x\right) t^\left\{n\right\}$

They are orthogonal with respect to the weighting function (Abramowitz & Stegun p774):

$w\left(z\right) = left\left(1-z^2right\right)^\left\{alpha-frac\left\{1\right\}\left\{2\right\}\right\}$

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