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.

Download

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},
}