A New Product Formula Involving Bessel Functions

Abstract: In this paper, we consider the normalized Bessel function of index $\alpha > -\frac{1}{2}$, we find an integral representation of the term $x^nj_{\alpha+n}(x)j_\alpha(y)$. This allows us to establish a product formula for the generalized Hankel function $B^{\kappa,n}_\lambda$ on $\mathbb{R}$. $B^{\kappa,n}_\lambda$ is the kernel of the integral transform $\mathcal{F}{\kappa,n}$ arising from the Dunkl theory. Indeed we show that $B^{\kappa,n}\lambda(x)B^{\kappa,n}_\lambda(y)$ can be expressed as an integral in terms of $B^{\kappa,n}_\lambda(z)$ 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 $B^{\kappa,n}_\lambda$. They share many important properties with their analogous in the classical Fourier theory.