Product formula for one-dimensional $(k,a)$-generalized Fourier kernel

TL;DR

This paper derives a product formula for the one-dimensional $(k,a)$-generalized Fourier kernel, extending previous special cases and broadening the theoretical understanding of these kernels for certain parameter ranges.

Contribution

It provides a new product formula for the $(k,a)$-generalized Fourier kernel applicable for $k\, ext{geq}\,0$, $a>0$, and $2k>a-1$, generalizing earlier specific cases.

Findings

Extended the product formula to a broader parameter range

Unified previous special cases under a general framework

Enhanced theoretical understanding of generalized Fourier kernels

Abstract

In this paper, a product formula for the one-dimensional $(k,a)$-generalized Fourier kernel is given for $k\geq0$, $a>0$ and $2k>a-1$, extending the special case when $a=\frac{2}{n}$, $n\in \mathbb{N}$.