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It is an open conjecture that generalized Bessel functions associated withroot systems have a positive product formula for non-negative multiplicityparameters of the associated Dunkl operators. In this paper, a partial resulttowards this conjecture is proven, namely a positive radial product formula forthe non-symmetric counterpart of the generalized Bessel function, the Dunklkernel. Radial hereby means that one of the factors in the product formula isreplaced by its mean over a sphere. The key to this product formula is apositivity result for the Dunkl-type spherical mean operator. It can also beinterpreted in the sense that the Dunkl-type generalized translation of radialfunctions is positivity-preserving. As an application, we construct Dunkl-typehomogeneous Markov processes associated with radial probability distributions.

A positive radial product formula for the Dunkl kernel