Generating Function of the Solution of a Difference Equation and the Newton Polyhedron of the Characteristic Polynomial

Abstract

Generating functions and difference equations are a powerful tool for studying problems of enumerative combinatorial analysis. In the one-dimensional case, the space of solutions of the difference equation is finite-dimensional. In the transition to a multidimensional situation, problems arise related both to the possibility of various options for specifying additional conditions on the solution of a difference equation (the Cauchy problem) and to describing the corresponding space of generating functions. For difference equations in rational cones of an integer lattice, sufficient conditions are known on the Newton polyhedron of the characteristic polynomial that ensure the preservation of the Stanley hierarchy for the generating functions of its solutions. Namely, a generating function is rational (algebraic, D-finite) if such are the generating functions of the initial data and the right side of the equation. In this paper, we propose an approach for finding the generating function of a solution to a difference equation based on the possibility of extending the rational cone in which solutions of the equation are sought to a cone in which sufficient conditions for the conservation of the Stanley hierarchy are satisfied. In addition, an integral formula is given that relates the generating functions of the solution in the original and extended cones.