On the operator ⊕k related to the wave equation and Laplacian

In this paper, we study the Green function of the operator @?^k, iterated k-times and is defined [email protected]?^[email protected]?"r"="1^[email protected]?^[email protected]?x^2"r^[email protected]?"j"="p"+"1^p^+^[email protected]?^[email protected]?x^2"j^4^k,where p+q=n is the dimension of the space C^n, where C is a complex field, x=(x"1,x"2,...,x"n)@?C^n and k is a nonnegative integer. At first we study the elementary solution or the Green function of the operator @?^k and then such a solution is related to the solution of the wave equation and the Laplacian. We found that the relationships of such solutions depending on the conditions of p, q and k.