Pre-sequences of matrix orthogonal polynomials

TL;DR

This paper introduces pre-sequences of matrix orthogonal polynomials, establishing a construction method from spherical functions and demonstrating their relation to matrix orthogonal polynomials via a specific transformation.

Contribution

It defines the concept of pre-sequences of matrix orthogonal polynomials and shows how to construct them from spherical functions associated with symmetric pairs.

Findings

Pre-sequences satisfy a three-term recursion relation.

The sequence Q_n = F_n F_0^{-1} forms matrix orthogonal polynomials.

The construction applies to compact symmetric pairs of rank one.

Abstract

We introduce the notion of a pre-sequence of matrix orthogonal polynomials to mean a sequence {F_n} 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}} is a sequence of matrix orthogonal polynomials with respect to the weight function F_0WF_0*, see Theorem 2.1.