## Symmetry of terminating series representations of the Askey-Wilson polynomials

Cohl, H. S. and Costas-Santos, R. S.**Journal of Mathematical analysis and applications**. Accepted in 2022

ACADEMIC PRESS INC ELSEVIER SCIENCE (USA) | ISSN: 0022-247X | JCR® 2021 Impact Factor: 1.417 - MATHEMATICS — position: 77/332 (Q1/T1)

## 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.

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