A New Product Formula Involving Bessel Functions

In this paper, we consider the normalized Bessel function of index $\alpha > -\frac{1}{2}$,wefindanintegralrepresentationoftheterm$x^nj_{\alpha+n}(x)j_\alpha(y)$. This allows us to establish a product formula for the generalized Hankel function $B_{\lambda }^{\kappa ,n}$ on $\mathbb{R}$. $B_{\lambda }^{\kappa ,n}$ is the kernel of the integral transform ${\mathcal{F}}_{\kappa ,n}$ arising from the Dunkl theory. Indeed we show that $B_{\lambda }^{\kappa ,n}(x)B_{\lambda }^{\kappa ,n}(y)$ can be expressed as an integral in terms of $B_{\lambda }^{\kappa ,n}(z)$ with explicit kernel invoking Gegenbauer polynomials for all $n\in {\mathbb{N}}^{\ast }$. The obtained result generalizes the product formulas proved by M. R"osler for Dunkl kernel when n=1 and by S. Ben Said when $n=2$. \ As application, we define and study a translation operator and a convolution structure associated to $B_{\lambda }^{\kappa ,n}$. They share many important properties with their analogous in the classical Fourier theory.