A new product formula involving Bessel functions
Mohamed Amine Boubatra, Selma Negzaoui, Mohamed Sifi
Abstract
In this paper, we consider the normalized Bessel function of index α>−12, we find an integral representation of the term xnjα+n(x)jα(y). This allows us to establish a product formula for the generalized Hankel function Bλκ,n on R. Bλκ,n is the kernel of the integral transform Fκ,n arising from the Dunkl theory. Indeed we show that Bλκ,n(x)Bλκ,n(y) can be expressed as an integral in terms of Bλκ,n(z) with explicit kernel invoking Gegenbauer polynomials for all n∈N∗. 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 Bλκ,n. They share many important properties with their analogous in the classical Fourier theory.