Darboux equivalence for matrix-valued orthogonal 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.