Three families of $q$-Lommel polynomials

TL;DR

This paper explores three $q$-Lommel polynomial families, providing explicit formulas, recurrences, continued fractions, and combinatorial interpretations, while connecting them to Askey--Wilson polynomials and moments.

Contribution

It introduces three new $q$-Lommel polynomial families with explicit representations and combinatorial insights, linking their moments to orthogonal polynomial theory.

Findings

Explicit formulas and recurrences for the $q$-Lommel polynomials

Connections established between $q$-Lommel polynomials and Askey--Wilson polynomials

Combinatorial interpretations using Motzkin paths, Schr"oder paths, and parallelogram polyominoes

Abstract

Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a general theorem when $R_I$ moments coincide with orthogonal polynomial moments. The combinatorial results use weighted Motzkin paths, Schr\"oder paths, and parallelogram polyominoes.