Darboux equivalence for matrix-valued orthogonal polynomials

TL;DR

This paper establishes criteria for when matrix-valued orthogonal polynomials are related through Darboux transformations, enabling explicit construction and analysis of their relationships to classical orthogonal polynomials.

Contribution

It introduces new criteria for identifying Darboux transformations between matrix-valued orthogonal polynomials and explores Darboux-irreducibility.

Findings

Criteria for Darboux relations between matrix-valued orthogonal polynomials

Explicit construction of Darboux transformations

Identification of sequences not related to classical polynomials

Abstract

In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see when and how any given sequence of polynomials is Darboux related to a diagonal matrix of classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.