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In this paper, we establish an integral expression for the Dunkl kernel inthe 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 alarge set of regular parameter functions is provided. We introduce adifferential equations systems that leads to the explicit expression of theDunkl 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 itsargument $x$ or $y$ is invariant under the action of a known reflection in thedihedral group. We obtain also a generating series for the homogeneouscomponents $E_m(x,y)$, $m\in\mathbb{Z}^{+}$, of the Dunkl kernel from which wederive 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.

An Integral Expression for the Dunkl Kernel in the Dihedral Setting