  In this paper, we derive explicit product formulas and positive convolution
structures for three continuous classes of Heckman-Opdam hypergeometric
functions of type $BC$. For specific discrete series of multiplicities these
hypergeometric functions occur as the spherical functions of non-compact
Grassmann manifolds $G/K$ over one of the (skew) fields $\mathbb F= \mathbb R,
\mathbb C, \mathbb H.$ We write the product formula of these spherical
functions in an explicit form which allows analytic continuation with respect
to the parameters. In each of the three cases, we obtain a series of hypergroup
algebras which include the commutative convolution algebras of $K$-biinvariant
functions on $G$.
