First order non-homogeneous q-difference equation for Stieltjes function characterizing q-orthogonal polynomials

Abstract

In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn are worked out.

A characterization of some q-classical polynomials (orthogonal with respect to some q-distributions) given in terms of a difference property of the associated Stieltjes function is presented. Explicit expressions for the associated Stieltjes functions (for some canonical cases) in the q-falling factorial basis are given in terms of hypergeometric functions