A reformulation of the generalized $q$-Painlev\'{e} VI system with $W(A^{(1)}_{2n+1})$ symmetry

TL;DR

This paper revises a higher order $q$-Painlevé system with $W(A^{(1)}_{2n+1})$ symmetry, making it a more suitable generalization of the classical $q$-Painlevé VI equation derived from affine Lie algebra hierarchies.

Contribution

The authors reformulate the higher order $q$-Painlevé system to better align with the classical $q$-Painlevé VI, enhancing its mathematical structure and potential applications.

Findings

Reformulated the $q$-$P_{(n+1,n+1)}$ system for better generalization.

Connected the system to affine Lie algebra $A^{(1)}_{2n+1}$ hierarchy.

Provided a more suitable framework for future analysis.

Abstract

In the previous work we introduced the higher order $q$-Painlev\'{e} system $q$-$P_{(n+1,n+1)}$ as a generalization of the Jimbo-Sakai's $q$-Painlev\'{e} VI equation. It is derived from a $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A^{(1)}_{2n+1}$ and admits a particular solution in terms of the Heine's $q$-hypergeometric function ${}_{n+1}\phi_n$. However the obtained system is insufficient as a generalization of $q$-$P_{\rm{VI}}$ due to some reasons. In this article we rewrite the system $q$-$P_{(n+1,n+1)}$ to a more suitable one.