Two variable Freud orthogonal polynomials and matrix Painlev\'e-type difference equations

TL;DR

This paper investigates bivariate Freud orthogonal polynomials, deriving matrix relations, structure relations, and extending Painlevé equations to a two-variable setting, revealing new integrable structures.

Contribution

It introduces a novel extension of Painlevé equations and Langmuir lattice to bivariate orthogonal polynomials associated with Freud weights.

Findings

Derived matrix coefficients for three-term relations

Established structure relations for bivariate polynomials

Extended Painlevé equations to two-variable case

Abstract

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyze relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlev\'e equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmuir lattice are obtained.