Gegenbauer polynomials

Summary

The Gegenbauer polynomials $C_n^{(\lambda)}$, also called ultraspherical polynomials, are orthogonal on $[-1, 1]$ for the weight $(1 - x^2)^{\lambda - 1/2}$. Up to constant factors they are the Jacobi polynomials with $\alpha = \beta = \lambda - \frac12$. They generalize the Legendre polynomials ($\lambda = \frac12$) and give the zonal spherical harmonics on spheres of every dimension at least $2$.