Explicit formulas for the Dunkl dihedral kernel and the $(\kappa, a)$-generalized Fourier kernel

TL;DR

This paper develops a new method to derive explicit integral formulas for the Dunkl dihedral kernel and the generalized Fourier kernel, utilizing Laplace transforms and Mittag-Leffler functions, with special cases for integer parameters.

Contribution

Introduces a novel approach using auxiliary variables and Laplace transforms to obtain explicit formulas for the Dunkl and generalized Fourier kernels, including bounds and special cases.

Findings

Explicit integral formulas for the Dunkl dihedral kernel.

New bounds for the $( )$-generalized Fourier kernel.

Formulas for integer parameters using partial fractions.

Abstract

In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the $(\kappa, a)$-generalized Fourier transform for $\kappa =0$. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag-Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the $(\kappa, a)$-generalized Fourier transform are obtained as well.