Multiple Laurent series and fundamental solutions of linear difference equations

Abstract

Using the notion of fundamental solution, we obtain a solution to the Cauchy problem for a multidimensional homogeneous linear difference equation with constant coefficients.Denote by Z the set of integers and by Z n = Z × · · · × Z the n-dimensional integer-valued lattice. Let Z n + be the subset of Z n of the points with nonnegative integer coordinates, and fix some finite subsetwhere c α are the (constant) coefficients of the equation.The characteristic polynomial of (1) is the polynomial α∈A c α z α =:The characteristic set of (1) is the set V = {z ∈ C n : P (z) = 0} of zeros of the characteristic polynomial.It is well known (for example, see [1]) that for n = 1 every solution to (1) is a linear combination of solutions of the form x s λ x , s = 0, . . . , k − 1, where λ ∈ V is a root of multiplicity k. Consequently, the solution space of (1) is finite-dimensional and its dimension equals the degree m of P (the order of the equation).Moreover, it is obvious that every solution to (1) is determined completely by the values f (x) = φ x at the "initial" points x = 0, 1, . . . , m − 1.For n > 1 the situation is essentially more complicated, since the solution space is infinite-dimensional and the question about the set on which the "initial" values of solutions should be given is not so obvious (neither the question of definition of the order of the difference equation). Observe that multidimensional difference equations (recurrent relations) arise in combinatorial analysis [2] and in discretization of differential equations (for example, see [3]).In [4] the author gave a description of the solution space of (1) using the notions of the Newton polyhedron and amoeba of the characteristic polynomial. In this article we define the set Z n m on which the "initial" values are given, state the Cauchy problem for (1), and using the notion of fundamental solution solve this problem.The Newton polyhedron N P of the characteristic polynomial P (z) is the convex hull in R n of the elements of A.Fix m ∈ N P ∩ Z n and denote Z n m j = {y ∈ Z n : 0 ≤ y j < m j }. Now, put n j=1 Z n m j =: Z n m , and let φ : Z n m → C be a given function.Problem. Find a solution to (1) coincident with φ on Z n m :f (x) = φ(x), x ∈ Z n m .(2)

The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones

“…The solvability of the problem when the cone K is simplicial (which means that every element in it admits a unique expansion in the generators) and the sets X = K and X 0 = X \ (m + K), on which Cauchy problem (1)-(2) is solved, was studied in [1,[10][11][12][13]19]. Additionally, in these papers, the solutions f (x) to problem (1)-(2) are given in terms of the Cauchy data and fundamental solution to (1)-(2) (the Green function).…”

8. On the Cauchy Problem for Multidimensional Difference Equations in Rational Cone

“…This element is equal to one and it lies on the main diagonal. This follows from the algorithm of ordering of unknowns and equations of system (5)- (6). Consider the rows of the determinant corresponding to equations (5).…”

“…Let us note that multisection is used to prove identities with binomial coefficients and the Bernoulli numbers [5]. The need for a multi-dimensional analogue of the notion of multisection multiple series arises in the study of the Cauchy problem for multidimensional difference equations (see [1,[6][7][8][9]). In particular, this is the case when supports of generating functions of equation solutions is in rational cone.…”

Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients

“…In the multidimensional case, which is not adequately explored (see [5,10,11,13]), the rational generating functions are the most useful class of generating functions according to the Stanley's hierarchy (see [19]). A broad class of twodimensional sequences that lead to rational generating functions is well-known in the enumerative combinatorics.…”