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

TL;DR

This paper characterizes classical q-orthogonal polynomials through a first order non-homogeneous q-difference equation satisfied by their Stieltjes functions, providing explicit solutions for key cases like q-Charlier and q-Hahn.

Contribution

It introduces a novel difference equation characterization for q-orthogonal polynomials and explicitly solves it for several canonical cases.

Findings

Solutions expressed in hypergeometric series

Characterization applies to q-Charlier, q-Kravchuk, q-Meixner, q-Hahn

Provides new insights into the structure of 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.