Spectral decomposition and matrix-valued orthogonal polynomials

TL;DR

This paper explores the connection between spectral decomposition of certain operators and matrix-valued orthogonal polynomials, providing a construction method and explicit examples related to Askey-Wilson polynomials.

Contribution

It introduces a general construction of operators from scalar orthogonal polynomials and presents explicit examples of matrix-valued orthogonal polynomials with orthogonality and recurrence relations.

Findings

Explicit examples of 2x2 matrix-valued orthogonal polynomials

Construction method linking scalar polynomials to matrix-valued cases

Orthogonality relations and recurrence relations derived

Abstract

The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as $2\times 2$-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.