An Integral Expression for the Dunkl Kernel in the Dihedral Setting (1505.05534v2)

Abstract: In this paper, we establish an integral expression for the Dunkl kernel in the context of Dihedral group of an arbitrary order by using the results in \cite{M-Y-Vk} where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided. We introduce a differential equations systems that leads to the explicit expression of the Dunkl Kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel $E_k(x,y)$ is given when one of its argument $x$ or $y$ is invariant under the action of a known reflection in the dihedral group. We obtain also a generating series for the homogeneous components $E_m(x,y)$, $m\in\mathbb{Z}^{+}$, of the Dunkl kernel from which we derive new sharp estimates for the Dunkl kernel when the parameter function $k$ satisfies $\mathrm{Re}(k)>-\nu$, $\nu$ an arbitrary nonnegative integer, which, up to our knowledge, is the largest context for such estimates so far.