Pre-sequences of matrix orthogonal polynomials
Juan Tirao

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.
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Taxonomy
TopicsMathematical functions and polynomials · Matrix Theory and Algorithms · Mathematical Approximation and Integration
