On the Elementary Solution for the Partial Differential Operator $\circledcirc_c^{k}$ Related to the Wave Equation

Abstract

In this article, we defined the operator $\diamondsuit _{m,c}^{k}$ which is iterated $k$-times and is defined by
$$\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{p}\frac{\partial ^{2}}{\partial x_{i}^{2}} +\frac{m^{2}}{2}\right)^{2} - \left(\sum_{j=p+1}^{p+q}\frac{\partial ^{2}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},$$
where $m$ is a nonnegative real number, $c$ is a positive real number and $p+q=n$ is the dimension of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, $x=(x_{1},\ldots x_{n})\in\mathbb{R}^{n}$
and $k$ is a nonnegative integer. We obtain a causal and anticausal solution
of the operator $\diamondsuit _{m,c}^{k}$, iterated $k$-times.