Paper: Symmetry of terminating series representations of the Askey-Wilson polynomials

Symmetry of terminating series representations of the Askey-Wilson polynomials

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},
}