Sufficient conditions of algebraicity of generating functions of the solutions of multidimensional difference equations

In this paper we study difference equations (recurrence relations) in rational lattice cones. It is found the relation between generating functions of the initial data and generating functions of the solutions of Cauchy problem for multidimensional difference equations. It is proved that condition of algebraicity (rationality) of generating functions of the initial data is sufficient condition of algebraicity (rationality) of generating functions of the solution.