A New Product Formula Involving Bessel Functions

Abstract

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