  The aim of this paper is to determine all irreducible spherical functions of
the pair (G,K)=(SU(n+1),U(n)), where the highest weight of their K-types are of
the form (m+l,...,m+l,m,...,m). Instead of looking at a spherical function \Phi
of type \pi we look at a matrix-valued function H defined on a section of the
K-orbits in an affine subvariety of P_n(C). The function H diagonalizes, hence
it can be identified with a column vector-valued function. The irreducible
spherical functions of type \pi turn out to be parameterized by S={(w,r)\in Z x
Z : 0\leq w, 0 \leq r \leq l, 0\leq m+w+r}. A key result to characterize the
associated function H_{w,r} is the existence of a matrix-valued polynomial
function \Psi of degree l such that F_{w,r}(t)=\Psi(t)^{-1}H_{w,r}(t) becomes
an eigenfunction of a matrix hypergeometric operator with eigenvalue
\lambda(w,r), explicitly given. In the last section we assume that m\ge 0 and
define the matrix polynomial P_w as the (l+1) x (l+1) matrix whose r-row is the
polynomial F_{w,r}. This leads to interesting families of matrix-valued
orthogonal Jacobi polynomials P_w^{\alpha,\beta} for \alpha,\beta>-1.
