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

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.

26 pages

References in corpus (1)

• The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C)

Cited by in corpus (5)

• Additions to the formula lists in "Hypergeometric orthogonal polynomials and their $q$-analogues" by Koekoek, Lesky and Swarttouw
• Orthogonality relations for Al-Salam--Carlitz polynomials of type II
• Macdonald-level extension of beta ensembles and large-N limit transition
• Coupling coefficients for tensor product representations of quantum $\mathrm{SU}(2)$
• A solution to the Al-Salam--Chihara moment problem