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