On the Elementary Solution for the Partial Differential Operator ⊚ck Related to the Wave Equation

Abstract

In this article, we defined the operator ${\mathrm{♢}}_{m,c}^k$ which is iterated $k$-times and is defined by
$${\mathrm{♢}}_{m,c}^k={[{(\frac{1}{c^2}\sum _{i=1}^p\frac{{\partial }^2}{\partial x_i^2}+\frac{m^2}{2})}^2-{(\sum _{j=p+1}^{p+q}\frac{{\partial }^2}{\partial x_j^2}-\frac{m^2}{2})}^2]}^k,$$
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 ${\mathbb{R}}^n$, $x=(x_1,\dots x_n)\in {\mathbb{R}}^n$
and $k$ is a nonnegative integer. We obtain a causal and anticausal solution
of the operator ${\mathrm{♢}}_{m,c}^k$, iterated $k$-times.