The Cauchy problem for multidimensional difference equations in lattice cones

We consider a variant of the Cauchy problem for a multidimensional difference equation with constant coefficients, which connected with a lattice path problem in enumerative combinatorial analysis. We obtained a formula in which generating function of the solution to the Cauchy problem is expressed in terms of generating functions of the Cauchy data and a formula expressing solution to the Cauchy problem through its fundamental solution and Cauchy data.