论文标题
涉及贝塞尔功能的新产品公式
A New Product Formula Involving Bessel Functions
论文作者
论文摘要
在本文中，我们考虑了索引$α> - \ frac {1} {2} $的归一化贝塞尔函数，我们找到了$ x^nj_ {α+n}（x）j_α（y）$的术语的积分表示。这使我们可以在$ \ mathbb {r} $上为广义Hankel函数$ b^{κ，n}_λ$建立产品公式。 $ b^{κ，n}_λ$是整体变换$ \ MATHCAL {f} _ {κ，n} $的内核。实际上，我们表明$ b^{κ，n}_λ（x）b^{κ，n}_λ（y）$可以用$ b^{κ，n}_λ（z）$表示为一个积分，并用明显的kernel intecokit necokit necokit necokit necokit necokit innebbbb in $ n \ n $ n \ in \ in \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ bb {获得的结果概括了M.Rösler为dunkl内核证明的产品配方时，当n = 1时，S。Ben说$ n = 2 $。 \\作为应用程序，我们定义并研究了与$ b^{κ，n}_λ$相关的翻译操作员和卷积结构。他们在经典傅立叶理论中具有许多重要的特性。
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.
