A new product formula involving Bessel functions

Abstract

In this paper, we consider the normalized Bessel function of index , we find an integral representation of the term . This allows us to establish a product formula for the generalized Hankel function on . is the kernel of the integral transform arising from the Dunkl theory. Indeed we show that can be expressed as an integral in terms of with explicit kernel invoking Gegenbauer polynomials for all . The obtained result generalizes the product formulas proved by M. Rösler 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 . They share many important properties with their analogous in the classical Fourier theory.