  In this paper we present explicit product formulas for a continuous
two-parameter family of Heckman-Opdam hypergeometric functions of type BC on
Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related
to continuous one-parameter families of probability-preserving convolution
structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$
are constructed via product formulas for the spherical functions of the
symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset
convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall
obtain positive product formulas for a restricted parameter set only, while the
associated convolutions are always norm-decreasing. Our paper is related to
recent positive product formulas of R\"osler for three series of Heckman-Opdam
hypergeometric functions of type BC as well as to classical product formulas
for Jacobi functions of Koornwinder and Trimeche for rank $q=1$.
