  We present an explicit product formula for the spherical functions of the
compact Gelfand pairs $(G,K_1)= (SU(p+q), SU(p)\times SU(q))$ with $p\ge 2q$,
which can be considered as the elementary spherical functions of
one-dimensional $K$-type for the Hermitian symmetric spaces $G/K$ with $K=
S(U(p)\times U(q))$. Due to results of Heckman, they can be expressed in terms
of Heckman-Opdam Jacobi polynomials of type $BC_q$ with specific half-integer
multiplicities. By analytic continuation with respect to the multiplicity
parameters we obtain positive product formulas for the extensions of these
spherical functions as well as associated compact and commutative hypergroup
structures parametrized by real $p\in]2q-1,\infty[$. We also obtain explicit
product formulas for the involved continuous two-parameter family of
Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive
multiplicities. The results of this paper extend well known results for the
disk convolutions for $q=1$ to higher rank.
