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

In this paper, we derive 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 by the second named author 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.