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

### Erik Koelink

Radboud Universiteit Nijmegen, Netherlands### Maarten van Pruijssen

Radboud Universiteit Nijmegen, Netherlands### Pablo Román

Universidad Nacional de Córdoba, Argentina

## Abstract

In a previous paper we have introduced matrix-valued 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. 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)$×$ SU(2).

## Cite this article

Erik Koelink, Maarten van Pruijssen, Pablo Román, Matrix-valued Orthogonal Polynomials Related to (SU(2)$×$ SU(2), diag), II. Publ. Res. Inst. Math. Sci. 49 (2013), no. 2, pp. 271–312

DOI 10.4171/PRIMS/106