The indeterminate moment problem for the $q$-Meixner polynomials

TL;DR

This paper investigates the indeterminate moment problem for a class of $q$-Meixner related orthogonal polynomials, providing explicit orthogonality measures and basis constructions through spectral and series methods.

Contribution

It introduces a one-parameter family of orthogonality measures for these polynomials and explicitly constructs the orthogonal basis in the weighted $L^2$-space.

Findings

Explicit orthogonality measures for the polynomials

Construction of orthogonal basis in $L^2$-space

Connections to other indeterminate moment problems

Abstract

For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous $q^{-1}$-Hahn and $q$-Laguerre polynomials.