Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

TL;DR

This paper develops a Riemann-Hilbert framework for matrix orthogonal polynomials, revealing their complex differential and algebraic properties, including new ladder operators and differential equations not present in scalar cases.

Contribution

It introduces a novel Riemann-Hilbert approach to matrix orthogonal polynomials, uncovering additional algebraic structures and differential relations unique to the matrix case.

Findings

Existence of ladder operators of 0-th order in matrix case

Derivation of second-order differential equations for matrix polynomials

Matrix case exhibits more complex differential properties than scalar case

Abstract

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.