Classical orthogonal polynomials

Summary

The classical orthogonal polynomials are the three families of Jacobi, Laguerre and Hermite polynomials. They include special cases such as the Gegenbauer, Legendre and Chebyshev polynomials. Each family is orthogonal for a weight on an interval. The weights are of beta type on $(-1, 1)$, of gamma type on $(0, \infty)$, and Gaussian on $\R$. They are the only orthogonal polynomial families that are eigenfunctions of a second-order differential operator, and they have Rodrigues formulas and explicit recurrences. They are the basic tools of approximation, Gaussian quadrature, and separation of variables in mathematical physics.