The Nevanlinna parametrization for $q$-Lommel polynomials in the indeterminate case

TL;DR

This paper explicitly describes the Nevanlinna parametrization for the indeterminate Hamburger moment problem associated with $q$-Lommel polynomials, providing a comprehensive characterization of all N-extremal measures and analyzing moment asymptotics.

Contribution

It offers an explicit Nevanlinna parametrization for the indeterminate case of $q$-Lommel polynomials, enabling complete description of orthogonality measures.

Findings

Explicit form of Nevanlinna parametrization for $q$-Lommel polynomials

Derivation of linear and quadratic recurrence relations for moments

Asymptotic analysis of moments for large powers

Abstract

The Hamburger moment problem for the $q$-Lommel polynomials which are related to the Hahn-Exton $q$-Bessel function is known to be indeterminate for a certain range of parameters. In this paper, the Nevanlinna parametrization for the indeterminate case is provided in an explicit form. This makes it possible to describe all N-extremal measures of orthogonality. Moreover, a linear and quadratic recurrence relation are derived for the moment sequence, and the asymptotic behavior of the moments for large powers is obtained with the aid of appropriate estimates.