Projective reduction of the discrete Painlev\'e system of type $(A_2+A_1)^{(1)}$

TL;DR

This paper investigates the reduction of the q-Painlevé III equation to q-Painlevé II, clarifying the apparent inconsistency in hypergeometric solutions through affine Weyl group symmetry, difference operator factorization, and tau functions.

Contribution

It introduces a new perspective on the reduction process of discrete Painlevé equations using affine Weyl group symmetry and operator factorization.

Findings

Clarifies the mechanism behind the inconsistency in hypergeometric solutions.

Provides a factorization approach to difference operators.

Elucidates the role of tau functions in the reduction process.

Abstract

We consider the q-Painlev\'e III equation arising from the birational representation of the affine Weyl group of type $(A_2 + A_1)^{(1)}$. We study the reduction of the q-Painlev\'e III equation to the q-Painlev\'e II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the $\tau$ functions.