Sections of the generating series of a solution to a difference equation in a simplicial cone

We consider a multidimensional difference equation in a simplicial lattice cone with coefficients from a field of characteristic zero and sections of a generating series of a solution to the Cauchy problem for such equations. We use properties of the shift and projection operators on the integer lattice $\mathbb Z^n$ to find a recurrence relation (difference equation with polynomial coefficients) for the section of the generating series. This formula allows us to find a generating series of a solution to the Cauchy problem in the lattice cone through a generating series of its initial data and a right-side function of the difference equation. We derived an integral representation for sections of the holomorphic function, whose coefficients satisfy the difference equation with complex coefficients. Finally, we propose a system of differential equations for sections that represent D-finite functions of two complex variables.