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Gegenbauer Polynomials
Definition
The Gegenbauer polynomials are defined as
\[
\begin{align}
C_n^{(\alpha)}(x) &= \frac{1}{\Gamma(\alpha)}\,\sum_{k=0}^{[n/2]} \frac{(-1)^k\,\Gamma(\alpha+n-k)}{k!\,(n-2k)!}\,(2x)^{n-2k}\\
&= {n+\alpha-1 \choose n} (2x)^n {}_2F_1 \left(\left. {-n/2, -n/2+\frac{1}{2} \atop -n-\alpha+1} \; \right| \frac{1}{x^2} \right)
\end{align}
\]
