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

Cohl, H. S. and Costas-Santos, R. S.**Journal of Mathematical analysis and applications**517, no. 1 (2023), 15 pages

## 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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## BibTeX

@article {MR4474848, AUTHOR={Cohl, Howard S. and Costas-Santos, Roberto S.}, TITLE={Symmetry of terminating basic hypergeometric series representations of the {A}skey-{W}ilson polynomials}, JOURNAL={J. Math. Anal. Appl.}, FJOURNAL={Journal of Mathematical Analysis and Applications}, VOLUME={517}, YEAR={2023}, NUMBER={1}, PAGES={Paper No. 126583, 15}, ISSN={0022-247X}, MRCLASS={33D15}, MRNUMBER={4474848}, DOI={10.1016/j.jmaa.2022.126583}, URL={https://doi.org/10.1016/j.jmaa.2022.126583}, }