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 $\left[\mathrm{-1}\,1\right]$ 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:

When $a=0$, $G\left(0\&comma;0\&comma;x\right)=1$, and $G\left(n\&comma;0\&comma;x\right)=\frac{2T\left(n\&comma;x\right)}{n}$ for $0<n$, where $T\left(n\&comma;x\right)$ is the nth Chebyshev polynomial of the first kind.

The ultraspherical polynomials satisfy the following recurrence relation, for $a\ne 0$.

$\mathrm{with}\left(\mathrm{orthopoly}\right)\&colon;$

$G\left(2\&comma;5\&comma;x\right)$

$60x^2-5$

$G\left(3\&comma;2\&comma;x\right)$

$32x^3-12x$

$G\left(17\&comma;-\frac{1}{3}\&comma;\frac{2}{3}\right)$

$\frac{64983745495942460}{12157665459056928801}$