Lax forms of the $q$-Painlev\'e equations

TL;DR

This paper presents Lax formalism representations for all $q$-Painlevé equations derived from the $q$-analog of the sixth Painlevé equation, highlighting their linear $q$-difference equation structures and degeneration patterns.

Contribution

It provides a unified Lax formalism for $q$-Painlevé equations and details their degeneration from type $A_2$, advancing understanding of their linear structures.

Findings

Lax formalism for all $q$-Painlevé equations derived from $q$-P6

Characterization of equations by associated linear $q$-difference data

Description of degeneration pattern from type $A_2$

Abstract

All $q$-Painlev\'e equations which are obtained from the $q$-analog of the sixth Painlev\'e equation are expressed in a Lax formalism. They are characterized by the data of the associated linear $q$-difference equations. The degeneration pattern from the $q$-Painlev\'e equation of type $A_2$ is also presented.