Differential properties of matrix orthogonal polynomials

TL;DR

This paper develops a comprehensive theory for semi-classical matrix orthogonal polynomials, characterizing their properties through distributional equations, quasi-orthogonality, structure relations, and differential equations, with illustrative examples.

Contribution

It introduces a general framework for semi-classical matrix orthogonal polynomials using distributional equations and characterizes their properties in a unified way.

Findings

Characterization of semi-classical matrix orthogonal polynomials via distributional equations

Derivation of quasi-orthogonality and structure relations for these polynomials

Examples illustrating the theoretical results

Abstract

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding (left) matrix orthogonal polynomials sequence. They involve a quasi-orthogonality property for their derivatives, a structure relation and a second order differo-differential equation. Finally we illustrate the preceding results with some non-trivial examples.