Symmetry of terminating series representations of the Askey-Wilson polynomials

TL;DR

This paper investigates the symmetric properties of terminating basic hypergeometric series related to Askey-Wilson polynomials, classifies equivalence classes, and explores their connection to symmetry groups and classical theorems.

Contribution

It classifies the symmetry and equivalence classes of terminating hypergeometric series associated with Askey-Wilson polynomials and links these to symmetry groups and classical hypergeometric identities.

Findings

Identified 4 and 7 equivalence classes of terminating basic hypergeometric series.

Connected these classes to the symmetry group S_6.

Provided a broader interpretation of Watson's q-analog of Whipple's theorem.

Abstract

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ${}_4\phi_3$ and terminating very-well poised ${}_8W_7$ basic hypergeometric series which are connected with the Askey--Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group $S_6$, the symmetry group of the terminating balanced ${}_4\phi_3$. We then use terminating balanced ${}_4\phi_3$ and terminating very-well poised ${}_8W_7$ transformations to give a broader interpretation of Watson's $q$-analog of Whipple's theorem and its converse. We give a broad description of the symmetry structure of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials.