  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.
