Let $\alpha > -1$ and $\beta > -1$. The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials with weight function $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ 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: █

Theorem: (Orthogonality) The following formula holds: $$\displaystyle\int_{-1}^1 (-1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)P_m^{(\alpha,\beta)}(x)dx=\dfrac{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n! \Gamma(n+\alpha+\beta+1)}\delta_{mn},$$ where $\delta_{mn}$ denotes the Dirac delta.

Proof: █

Theorem: The $P_n^{(\alpha,\beta)}$ functions satisfy the differential equation $$(1-x^2)\dfrac{d^2y}{dx^2}+(\beta-\alpha-(\alpha+\beta+2)x)\dfrac{dy}{dx}+n(n+\alpha+\beta+1)y=0.$$

Proof: █

Theorem

The following formula holds: $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ where $C_n$ denotes a Gegenbauer C polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a Jacobi P polynomial.