A $3 \times 3$ Lax Form for the $q$-Painlev\'e Equation of Type $E_6$

TL;DR

This paper introduces a new 3x3 matrix Lax form and a third order scalar equation for the $q$-Painlevé equation of type $E_6$, expanding the mathematical tools available for its analysis.

Contribution

The paper presents a novel 3x3 matrix Lax form and a third order scalar equation for the $q$-Painlevé $E_6$ equation, extending previous 2x2 and second order forms.

Findings

New 3x3 matrix Lax form for $E_6$ $q$-Painlevé

Third order scalar equation related to the Lax form

Discussion of the continuous limit of the equations

Abstract

For the $q$-Painlev\'e equation with affine Weyl group symmetry of type $E_6^{(1)}$, a $2\times 2$ matrix Lax form and a second order scalar lax form were known. We give a new $3\times 3$ matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed.