Symmetry of terminating series representations of the Askey-Wilson polynomials

Howard S. Cohl; Roberto S. Costas-Santos

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arXiv · 2202.08911

Symmetry of terminating series representations of the Askey-Wilson polynomials

Howard S. Cohl·Roberto S. Costas-Santos

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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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Howard S. Cohl, Roberto S. Costas-Santos. 2022-02-17. Symmetry of terminating series representations of the Askey-Wilson polynomials. https://arxiv.org/abs/2202.08911

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