Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II

Abstract

In a previous paper we have introduced matrixvalued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. In this paper, 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 operators of first and second order of 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 Pn. 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) × SU(2).