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Matrix Bessel Biorthogonal Polynomials: A Riemann-Hilbert approach

Amílcar Branquinho, Ana Foulquié-Moreno, Assil Fradi, Manuel Mañas

TL;DR

This work addresses matrix orthogonal polynomials of Bessel type defined via a matrix Pearson equation and develops a Riemann--Hilbert framework to encode the polynomials and their second-kind functions. It establishes RH problems for left and right MBPs, derives transfer matrices and zero-curvature relations, and connects these objects to differential equations through a Miura-like map, yielding novel scalar Bessel-type ODEs and a non-Abelian discrete Painlevé IV–type dynamics for the recursion coefficients in semiclassical cases. The results generalize scalar Bessel theory to the matrix setting, providing explicit semiclassical examples and revealing noncommutative Painlevé-type structures with potential applications to matrix-valued probability weights and approximation theory. Overall, the paper lays a robust RH-based foundation for matrix Bessel-type orthogonal polynomials and their integrable-difference dynamics.

Abstract

We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann-Hilbert problem we derive first and second order differential relations for the matrix orthogonal polynomials and functions of second kind. It is shown that the corresponding matrix recurrence coefficients satisfy a non-Abelian extensions of a family of discrete Painlevé d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.

Matrix Bessel Biorthogonal Polynomials: A Riemann-Hilbert approach

TL;DR

This work addresses matrix orthogonal polynomials of Bessel type defined via a matrix Pearson equation and develops a Riemann--Hilbert framework to encode the polynomials and their second-kind functions. It establishes RH problems for left and right MBPs, derives transfer matrices and zero-curvature relations, and connects these objects to differential equations through a Miura-like map, yielding novel scalar Bessel-type ODEs and a non-Abelian discrete Painlevé IV–type dynamics for the recursion coefficients in semiclassical cases. The results generalize scalar Bessel theory to the matrix setting, providing explicit semiclassical examples and revealing noncommutative Painlevé-type structures with potential applications to matrix-valued probability weights and approximation theory. Overall, the paper lays a robust RH-based foundation for matrix Bessel-type orthogonal polynomials and their integrable-difference dynamics.

Abstract

We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann-Hilbert problem we derive first and second order differential relations for the matrix orthogonal polynomials and functions of second kind. It is shown that the corresponding matrix recurrence coefficients satisfy a non-Abelian extensions of a family of discrete Painlevé d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.

Paper Structure

This paper contains 4 sections, 8 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Riemann--Hilbert Formulation for MBPs
  3. Differential Equations: Insights and Applications
  4. Application for semiclassical weights of Bessel type

Key Result

Theorem 1

Given a regular Bessel type matrix of weights $W$ with support on $\gamma$, given in Definition def:1, we have the matrix function $Y^{\mathsf L}_n$ and $Y^{\mathsf R}_n$, defined in eq:ynl are, for each $n \in \mathbb{N}$, the unique solution of the following Riemann--Hilbert problem, which consist

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 6 more