Hypergeometric $\tau$ Function of the $q$-Painlev\'e Systems of Type $(A_2+A_1)^{(1)}$

TL;DR

This paper investigates hypergeometric solutions of specific $q$-Painlevé equations associated with the affine Weyl group of type $(A_2+A_1)^{(1)}$, focusing on their $ au$ functions.

Contribution

It introduces a detailed analysis of hypergeometric $ au$ functions for $q$-Painlevé III and II equations linked to the $(A_2+A_1)^{(1)}$ affine Weyl group.

Findings

Explicit hypergeometric solutions for the $q$-Painlevé III and II equations.

Connection formulas for $ au$ functions in the context of affine Weyl groups.

Enhanced understanding of the structure of solutions in the $q$-Painlevé systems.

Abstract

We consider a $q$-Painlev\'e III equation and a $q$-Painlev\'e II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $\tau$ functions.