Matrix Valued Orthogonal Polynomials related to (SU(2) x SU(2),diag)

TL;DR

This paper develops matrix-valued orthogonal polynomials derived from spherical functions related to the pair (SU(2) x SU(2), diag), providing explicit formulas, recurrence relations, and differential operators.

Contribution

It introduces a new class of matrix-valued orthogonal polynomials from spherical functions, with explicit weight functions, block-diagonalization, and differential operators, expanding the theory of matrix-valued special functions.

Findings

Explicit formulas for the weight functions and orthogonal polynomials.

Derivation of three-term recurrence relations.

Identification of differential operators with these polynomials as eigenfunctions.

Abstract

The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions we obtain matrix-valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression we obtain right-hand sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues formula and being eigenfunctions to first order differential-difference and second order differential operators, are obtained.