On difference equations for orthogonal polynomials on nonuniform lattices1

Abstract

By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181-226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey-Wilson operator [image omitted], following pioneering work by Magnus 1988, Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261-278, connecting [image omitted] and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients.