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

Recently, we formulated the $q$-Garnier system and its variations as translations of an extended affine Weyl group of type $A_{2n+1}^{(1)}\times A_1^{(1)}\times A_1^{(1)}$. 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 ${}_{n+1}{\phi }_n$.