Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlev\'e System

TL;DR

This paper constructs a Lax pair for the $E_6^{(1)}$ $q$-Painlevé system using semi-classical orthogonal polynomial theory on quadratic lattices, providing new explicit equations and connections to existing Lax pairs.

Contribution

It introduces a novel Lax pair construction for the $E_6^{(1)}$ $q$-Painlevé system based on divided-difference operators and orthogonal polynomials, linking it to previous formulations.

Findings

Derived coupled first-order $q$-difference equations for the system.

Established explicit transformations linking to earlier Lax pairs.

Extended the theory of orthogonal polynomials on quadratic lattices.

Abstract

We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6 $ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.