New Identities for Orthogonal Polynomials on a Compact Interval

Abstract

We derive new identities for orthonormal polynomials with respect to an arbitrary (probability) measure on the interval [−1, 1], which generalize the well known identity (1 − x2) U2n−1(x) + T2n(x) = 1 for the Chebyshev polynomials of the first (Tn) and second kind (Un). The results are established using necessary conditions of statistical optimal design theory for weighted polynomial regression. As special cases new identities are given for the Legendre, Chebyshev, ultraspherical, and Jacobi polynomials.