Hypergeometric $\tau$ Functions of the $q$-Painlev\'e Systems of Types $A_4^{(1)}$ and $(A_1+A_1')^{(1)}$

TL;DR

This paper investigates hypergeometric solutions of specific $q$-Painlevé equations associated with affine Weyl groups of types $A_4^{(1)}$ and $(A_1+A_1')^{(1)}$, focusing on $ au$ functions.

Contribution

It introduces hypergeometric $ au$ functions for these $q$-Painlevé systems, providing new explicit solutions linked to their Weyl group symmetries.

Findings

Derived explicit hypergeometric solutions for $A_4^{(1)}$ and $(A_1+A_1')^{(1)}$ $q$-Painlevé equations.

Connected $ au$ functions with hypergeometric functions and Weyl group actions.

Enhanced understanding of special solutions in integrable $q$-difference equations.

Abstract

We consider $q$-Painlev\'e equations arising from birational representations of the extended affine Weyl groups of $A_4^{(1)}$- and $(A_1+A_1)^{(1)}$-types. We study their hypergeometric solutions on the level of $\tau$ functions.