Laplace-type integral representations of the generalized Bessel function and of the Dunkl kernel of type $B_2$

TL;DR

This paper derives Laplace-type integral representations for the generalized Bessel function and Dunkl kernel associated with the B_2 root system, connecting them to known special functions and measures.

Contribution

It provides new integral representations for these functions, extending previous results and applying the shift principle to derive the Dunkl kernel representation.

Findings

Integral representation of the generalized Bessel function using modified Bessel functions

Expression of the Duistermaat-Heckman measure density for the dihedral group of order eight

Laplace-type integral representation of the Dunkl kernel for type B_2

Abstract

In this paper, we derive a Laplace-type integral representations for both the generalized Bessel function and the Dunkl kernel associated with the rank-two root system of type B_2. The derivation of the first one elaborates on the integral representation of the generalized Bessel function proved in \cite{Demni} through the modified Bessel function of the first kind. In particular, we recover an expression of the density of the Duistermaat-Heckman measure for the dihedral group of order eight. As to the integral representation of the corresponding Dunkl kernel, it follows from an application of the shift principle to the generalized Bessel function.