Lax pairs of discrete Painlev\'e equations: $(A_2+A_1)^{(1)}$ case

TL;DR

This paper introduces a systematic method for constructing Lax pairs of discrete Painlevé equations, focusing on the $(A_2+A_1)^{(1)}$ case, and presents two newly discovered Lax pairs for the $A_5^{(1)}$-surface $q$-Painlevé system.

Contribution

The paper develops a comprehensive method using a reduced hypercube structure to construct Lax pairs for discrete Painlevé equations, specifically for the $(A_2+A_1)^{(1)}$ case, and reports two new Lax pairs.

Findings

Two new Lax pairs for the $A_5^{(1)}$-surface $q$-Painlevé system.

A systematic construction method using reduced hypercube structure.

Enhanced understanding of the affine Weyl group symmetry in discrete Painlevé equations.

Abstract

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlev\'e equations by using a reduced hypercube structure. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlev\'e system which has the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$. Two new Lax pairs are found.