An affine Weyl group action on the basic hypergeometric series arising from the $q$-Garnier system

TL;DR

This paper explores how an extended affine Weyl group acts on basic hypergeometric series related to the $q$-Garnier system, revealing new symmetries and structures in these special functions.

Contribution

It introduces an affine Weyl group action on the basic hypergeometric series ${}_{n+1} ext{phi}_n$, connecting group symmetries with hypergeometric solutions of the $q$-Garnier system.

Findings

Identified a new affine Weyl group symmetry of ${}_{n+1} ext{phi}_n$ series.

Linked the $q$-Garnier system solutions to affine Weyl group actions.

Enhanced understanding of symmetries in basic hypergeometric functions.

Abstract

Recently, we formulated the $q$-Garnier system and its variations as translations of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$. On the other hand, those systems admit particular solutions in terms of the basic hypergeometric series ${}_{n+1}\phi_n$. In this article, we investigate an action of the extended affine Weyl group on ${}_{n+1}\phi_n$.