Matrix Bessel Biorthogonal Polynomials: A Riemann-Hilbert approach

TL;DR

This paper develops a Riemann-Hilbert framework for matrix Bessel biorthogonal polynomials, deriving differential relations and non-Abelian Painlevé equations, with examples illustrating their properties.

Contribution

It introduces a novel Riemann-Hilbert approach to matrix Bessel polynomials and links their recurrence coefficients to non-Abelian Painlevé equations.

Findings

Derived differential relations for matrix orthogonal polynomials.

Established non-Abelian discrete Painlevé equations for recurrence coefficients.

Provided explicit examples of matrix Bessel-type orthogonal polynomials.

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 Painlev\'e d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.