On q-Hermite polynomials and their relationship with some other families of orthogonal polynomials

TL;DR

This paper reviews the properties of q-Hermite polynomials and explores their connections with other orthogonal polynomial families, providing a comprehensive collection of known facts and relationships.

Contribution

It consolidates scattered literature on q-Hermite polynomials and details their links with Chebyshev, Rogers--Szeg"o, Al-Salam--Chihara, and continuous q-ultraspherical polynomials.

Findings

Connection coefficients between polynomial families are detailed.

Includes finite and infinite expansions involving these polynomials.

Provides a comprehensive literature review of known properties.

Abstract

We review properties of the $q-$Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szeg\"{o}, Al-Salam--Chihara, continuous $q-$% utraspherical polynomials. In particular we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.