  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
$G_{\lambda}^{(\alpha,\beta)}$.
