Jacobi polynomials

Remarks

Equivalently, $P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{n!} \, {}_2F_1\left( -n, n + \alpha + \beta + 1; \alpha + 1; \frac{1 - x}{2} \right)$. Rodrigues' formula: $(1 - x)^\alpha (1 + x)^\beta P_n^{(\alpha, \beta)}(x) = \frac{(-1)^n}{2^n n!} \frac{d^n}{dx^n} \left( (1 - x)^{\alpha + n} (1 + x)^{\beta + n} \right)$. They are orthogonal: $\int_{-1}^{1} P_m^{(\alpha, \beta)}(x) P_n^{(\alpha, \beta)}(x) (1 - x)^\alpha (1 + x)^\beta \, dx = 0$ for $m \ne n$. Example: $P_1^{(\alpha, \beta)}(x) = (\alpha + 1) + (\alpha + \beta + 2) \frac{x - 1}{2}$; for $\alpha = \beta = 0$ this is $x$, the Legendre polynomial $P_1$. The value at $1$ is $P_n^{(\alpha, \beta)}(1) = \frac{(\alpha + 1)_n}{n!}$, since only the term $k = 0$ survives. Non-example: for $\alpha = -1$ the weight $(1 - x)^{-1}$ is not integrable near $x = 1$, so the clause $\alpha > -1$ fails and the orthogonality integrals diverge.