A New Product Formula Involving Bessel Functions

TL;DR

This paper derives a new product formula for generalized Hankel functions related to Dunkl theory, extending previous results and enabling the development of associated translation and convolution operators with classical Fourier properties.

Contribution

It introduces a novel product formula for the kernel $B^{ ext{κ,n}}_λ$ involving Bessel functions, generalizing existing Dunkl kernel formulas for all natural numbers n.

Findings

Established an integral representation for $x^nj_{α+n}(x)j_α(y)$.

Derived a product formula expressing $B^{ ext{κ,n}}_λ(x)B^{ ext{κ,n}}_λ(y)$ as an integral involving $B^{ ext{κ,n}}_λ(z)$.

Developed translation and convolution operators with properties analogous to classical Fourier analysis.

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.