Pre-sequences of matrix orthogonal polynomials

Abstract

We introduce the notion of a pre-sequence of matrix orthogonal polynomials to mean a sequence $$\{F_n\}_{n\ge 0}$$ of matrix orthogonal functions with respect to a weight function W, satisfying a three term recursion relation and such that $$\det (F_{0})$$ is not zero almost everywhere. By now there is a uniform construction of such sequences from irreducible spherical functions of some fixed K-types associated to compact symmetric pairs (G, K) of rank one. Our main result is that $$\{Q_n=F_nF_0^{-1}\}_{n\ge 0}$$ is a sequence of matrix orthogonal polynomials with respect to the weight function $$F_0WF_0^*$$ , see Theorem 21.