On the Solvability of the Cauchy Problem for a Polynomial Difference Operator

In this paper we consider a variant of the Cauchy problem for a polynomial difference operator. A formula representing the solution to the problem via its fundamental solution is obtained, the conditions for its solvability are given. In particular, in two-dimensional case a rather simple condition in terms of coefficients of the operator symbol is proved. Besides that, a recurrence relation for principal minors of the matrix corresponding to the difference operator is obtained.