Matrix-valued Orthogonal Polynomials Related to (SU(2)$\times$ SU(2), diag), II

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 $P_{n}L$ 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).