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

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.