Riemann--Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement

TL;DR

This paper links matrix orthogonal polynomials to Riemann--Hilbert problems, deriving discrete matrix equations with singularity confinement, especially for matrix Freud cases, revealing new integrability properties.

Contribution

It introduces a Riemann--Hilbert framework for matrix orthogonal polynomials and derives explicit discrete equations with singularity confinement properties.

Findings

Discrete equations derived for matrix Freud polynomials

Singularity confinement shown under triangularizability conditions

Framework facilitates analysis of matrix orthogonal polynomial recursions

Abstract

In this paper matrix orthogonal polynomials in the real line are described in terms of a Riemann--Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by recursion coefficients to quartic Freud matrix orthogonal polynomials or not.