Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Let G(lambda)((alpha,beta)) be the eigenfunctions of the Dunkl-Cherednik operator T-(alpha,T-beta) on R. In this paper we express the product G(lambda)((alpha,beta))(x) G(lambda)((alpha,beta))(y) as an integral in terms of G(lambda)((alpha,beta))(z) with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rosler for the Dunkl kernel. We then define and study a convolution structure associated to G(lambda)((alpha,beta)).