The Cauchy problem for multidimensional difference equation with constant coefficients (Q1397513)

The author proves that the Cauchy problem for a multidimensional difference equation of the form \(\sum_{\alpha\in A}a_\alpha f(x+\alpha)=0\) with constant coefficients \(a_\alpha\), where \(A=\{\alpha\}\) is a finite set of nonnegative multi-indices and \(f: {\mathbb{N}}_0^n\to {\mathbb{C}}\); has a unique solution for a suitable set of initial conditions. Moreover the explicit expression of such solution is given.