  The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are
studied. By restriction to the subgroup A the matrix-valued spherical functions
are diagonal. For suitable set of representations we take these diagonals into
a matrix-valued function, which are the full spherical functions. Their
orthogonality is a consequence of the Schur orthogonality relations. From the
full spherical functions we obtain matrix-valued orthogonal polynomials of
arbitrary size, and they satisfy a three-term recurrence relation which follows
by considering tensor product decompositions. An explicit expression for the
weight and the complete block-diagonalization of the matrix-valued orthogonal
polynomials is obtained. From the explicit expression we obtain right-hand
sided differential operators of first and second order for which the
matrix-valued orthogonal polynomials are eigenfunctions. We study the
low-dimensional cases explicitly, and for these cases additional results, such
as the Rodrigues formula and being eigenfunctions to first order
differential-difference and second order differential operators, are obtained.
