On Correctness of Cauchy problem for a Polynomial Difference Operator with Constant Coefficients

The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones

“…In the general situation we have to solve the Cauchy problem: find a function f (x), satisfying equation (1) and coinciding with a given function φ(x) of initial data on the set X 0 :…”

“…For example, option (i) is used to describe the solution to equation (1) in the theory of discrete dynamic systems (see [6]). Option (ii) is most useful in problems of enumerative combinatorial analysis (see [17]).…”

“…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).…”

“…In the case of lattice paths, φ(y 1 , 0) − φ(y 1 − 1, 0) = 0 for y 1 1, φ(0, y 2 ) − φ(0, y 2 − 1) = 0 for y 2 1, and φ(0, 0) = 1, we obtain…”

An Approach to Multidimensional Discrete Generating Series

“…arises in a wide class of combinatorial analysis problems [3], for instance, in lattice path problems [4], the theory of digital recursive filters [14], and the wavelet theory [15]. The question about correctness and well-posedness of (2) was considered in [16][17][18]. We equip (1) with initial data on a set named X m , which is used often enough.…”