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orthopoly

 G
 Gegenbauer (ultraspherical) polynomial Calling Sequence G(n, a, x) Parameters

 n - non-negative integer a - rational number greater than $-\frac{1}{2}$ or nonrational algebraic expression x - algebraic expression Description

 • The G(n, a, x) function computes the nth ultraspherical (Gegenbauer) polynomial with parameter a evaluated at x.
 • These polynomials are orthogonal on the interval $\left[-1,1\right]$ with respect to the weight function $w\left(x\right)={\left(-{x}^{2}+1\right)}^{a-\frac{1}{2}}$. For $a\ne 0$ they satisfy:

${\int }_{-1}^{1}w\left(t\right)G\left(m,a,t\right)G\left(n,a,t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{\mathrm{\pi }{2}^{1-2a}\mathrm{\Gamma }\left(n+2a\right)}{n!\left(n+a\right){\mathrm{\Gamma }\left(a\right)}^{2}}& n=m\end{array}\right\$

 • When $a=0$, $G\left(0,0,x\right)=1$, and $G\left(n,0,x\right)=\frac{2T\left(n,x\right)}{n}$ for $0, where $T\left(n,x\right)$ is the nth Chebyshev polynomial of the first kind.
 • The ultraspherical polynomials satisfy the following recurrence relation, for $a\ne 0$.

$G\left(0,a,x\right)=1$

$G\left(1,a,x\right)=2ax$

$G\left(n,a,x\right)=\frac{2\left(n+a-1\right)xG\left(n-1,a,x\right)}{n}-\frac{\left(n+2a-2\right)G\left(n-2,a,x\right)}{n},\mathrm{for n>1.}$ Examples

 > $\mathrm{with}\left(\mathrm{orthopoly}\right):$
 > $G\left(2,5,x\right)$
 ${60}{}{{x}}^{{2}}{-}{5}$ (1)
 > $G\left(3,2,x\right)$
 ${32}{}{{x}}^{{3}}{-}{12}{}{x}$ (2)
 > $G\left(17,-\frac{1}{3},\frac{2}{3}\right)$
 $\frac{{64983745495942460}}{{12157665459056928801}}$ (3)