Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials

TL;DR

This paper explores the connection between Hermite-type matrix orthogonal polynomials, Riemann-Hilbert problems, and non-commutative Painleve' IV equations, revealing new integrable structures in matrix analysis.

Contribution

It introduces a non-commutative Painleve' IV equation derived from Riemann-Hilbert problems associated with matrix orthogonal polynomials.

Findings

Fredholm determinants relate to Riemann-Hilbert problems

Lax pairs lead to non-commutative Painleve' IV equations

New integrable structures in matrix orthogonal polynomials

Abstract

We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.