An affine Weyl group action on the basic hypergeometric series arising from the $q$-Garnier system (2207.03185v2)

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 ${}{n+1}\phi_n$.