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.