On correctness of Cauchy problem for a polynomial difference operator with constant coefficients

The theory of linear difference equations is applied in various areas of mathematics and in the one-dimensional case is quite established. For $n>1$, the situation is much more difficult and even for the constant coefficients a general description of the space of solutions of a difference equation is not available.
In the combinatorial analysis, difference equations combined with the method of generating functions produce a powerful tool for investigation of enumeration problems. Another instance when difference equations appear is the discretization of differential equations. In particular, the discretization of the Cauchy–Riemann equation led to the creation of the theory of discrete analytic functions which found applications in the theory of Riemann surfaces and the combinatorial analysis. The methods of discretization of a differential problem are an important part of the theory of difference schemes and also lead to difference equations. The existence and uniqueness of a solution is one of the main questions in the theory of difference schemes.
Another important question is the stability of a difference equation. For $n=1$ and constant coefficients the stability is investigated in the framework of the theory of discrete dynamical systems and is completely defined by the roots of the characteristic polynomial, namely: they all lie in the unit disk.
In the present work, we give two easily verified sufficient conditions on the coefficients of a difference operator which guarantee the correctness of a Cauchy problem.