Jacobi polynomials

Summary

The Jacobi polynomials $P_n^{(\alpha, \beta)}$ are the classical orthogonal polynomials on $[-1, 1]$ for the weight $(1 - x)^\alpha (1 + x)^\beta$, with $\alpha, \beta > -1$. They include the Gegenbauer polynomials, the Legendre polynomials and both kinds of Chebyshev polynomials as special cases, up to constant factors. They are terminating hypergeometric functions, and they appear in harmonic analysis on spheres and on compact symmetric spaces.