On the Study of the Klein-Gordon Equation in the Dunkl Setting

In Dunkl theory on ${\mathbb{R}}^n$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-Δ_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \ \ \partial_{t}u(x,0)=f(x) \end{eqnarray} with \ $m>0$ \ and \ ${\mathrm{\partial }}_t^{2}u$ \ is the second derivative of the solution $u$ with respect to $t$ and ${\mathrm{\Delta }}_{k}u$ is the Dunkl Laplacian with respect to $x$ where $f$ and $g$ the two functions in $\mathcal{S}({\mathbb{R}}^n)$ which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl-Klein-Gordon equation.