Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials

TL;DR

This paper thoroughly investigates the terminating hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric Askey--Wilson polynomial subfamilies, revealing their transformation structures and symmetry group properties.

Contribution

It provides a comprehensive analysis of terminating hypergeometric representations and transformations for specific Askey--Wilson polynomial subfamilies, highlighting their symmetry and structural properties.

Findings

Exhaustive hypergeometric representations of subfamilies

Transformation formulas derived from symmetry in parameters

Description of the symmetry group structure of the $q$-Askey scheme

Abstract

In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly setting one of the free parameters (not $q$) equal to zero until no parameters are left. These subfamilies (and their $q^{-1}$ counterparts) are the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite, and the continuous $q$-Hermite polynomials. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulas which these polynomials satisfy. We then study the terminating transformation structure which are implied by the terminating representations of these polynomials. We conclude by describing the symmetry group structure of the $q$-Askey scheme.