Matrix-valued orthogonal polynomials related to (SU(2)\times SU(2),diag), II

TL;DR

This paper develops and analyzes matrix-valued orthogonal polynomials related to the group SU(2)×SU(2), providing explicit formulas, decompositions, and group-theoretic derivations for these polynomials and their properties.

Contribution

It introduces explicit formulas and decompositions for matrix-valued orthogonal polynomials associated with SU(2)×SU(2), extending previous work to larger matrices and group-theoretic contexts.

Findings

LDU-decomposition of the weight function with Gegenbauer polynomials

Explicit three-term recurrence coefficients for P_n

Group-theoretic derivation of differential operators

Abstract

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P_n. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)\times SU(2).