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 are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials 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 's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials . These differential operators are also crucial in expressing the matrix entries of 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).
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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. 271–312DOI 10.4171/PRIMS/106