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 $T^{(\alpha ,\beta )}$ on $\mathbb{R}$. 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. R"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda }^{(\alpha ,\beta )}$.