On derivatives of orthogonal polynomials

Abstract

m{x)(j)n{x)d\p(x) = 0, I d\p(x) > 0, (m 7* n), a J a where \[/(x) is a non-decreasing function of bounded variation. There is no restriction in assuming the highest coefficient 1. It has been shown by W. Hahnf that if the derivatives also form a set of orthogonal polynomials, then the original set were Jacobi, Hermite, or Laguerre polynomials. His method consisted in showing that the polynomials satisfy a differential equation of the type (a + bx + cx)(j)n + (d + ex)4>n + \n<£n = 0.