Jacobi P
From specialfunctionswiki
The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are defined by $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ where ${}_2F_1$ is the generalized hypergeometries series.
Properties
Theorem: (Rodrigues' formula) The following formula holds: $$P_n^{(\alpha,\beta)}(z)=\dfrac{(-1)^n}{2^nn!} (1-z)^{-\alpha}(1+z)^{-\beta} \dfrac{d^n}{dz^n} \left[(1-z)^{\alpha}(1+z)^{\beta}(1-z^2)^n \right].$$
Proof: █
