Hypergeometric $\tau$-Functions of the $q$-Painlev\'e System of Type $E_7^{(1)}$

Abstract

We present the $\tau$-functions for the hypergeometric solutions to the $q$-Painlev\'e system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8W_7$. By using the $W(D_5)$ symmetry of the function ${}_8W_7$, we construct a set of twelve solutions and describe the action of $\widetilde{W}(D_6^{(1)})$ on the set.