Lax-type pairs in the theory of bivariate orthogonal polynomials

2023/11/10 by Amílcar Branquinho, Branquinho, Amílcar, Ana Foulquié‐Moreno +5
Mathematics · #33C50 #35Q53 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2311.05949

openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems.

Related

• Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle
• Quadratic decomposition of bivariate orthogonal polynomials
• Three term relations for a class of bivariate orthogonal polynomials
• Characterization of classical orthogonal polynomials in two variables
• Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations