Mathematics & Physics 2015, 8(2), 184–191 УДК 517.55 On the Cauchy Problem for Multidimensional Difference Equations in Rational Cone Tatiana I.Nekrasova∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.02.2015, received in revised form 15.03.2015, accepted 28.04.2015 The Cauchy problem for multidimensional difference equations in rational cone is formulated and sufficient condition for its solvability is given. <...> The notion of multisection of multiple Laurent series with the support in a rational cone is defined. <...> Introduction In this paper we discuss some issues related to the Cauchy problem for multidimensional difference equations whose solutions are sought at the intersection of rational cone K with integer lattice. <...> Methods of the theory of generating functions (z-transformations) play an important role in the study of the Cauchy problem. <...> Problems of solvability of the Cauchy problemin the positive octant of the integer lattice and the algebraic nature of the generating function of its solution are studied in [1]. <...> When passing from positive octant to more general case of a rational cone difficulties arise. <...> In the first section we formulate the Cauchy problem and provide the sufficient condition for its solvability (see Theorem 1). <...> The multi-dimensional analogue of the notion of the multisection of a power series helps us to overcomementioned above difficulties in study of generating functions (series) with supports in rational cones. <...> Relation that represents the multisections of the series in terms of the original series (see Theorem 2) is also presented in the second part of the paper. 1. <...> On solvability of the Cauchy problem Let us introduce complex-valued functions f(x) = f(x1, . . . ,xn) of integer variables x1, . . . ,xn. <...> We define the shift operators δj with respect to the variables xj: δjf(x) = f(x1, . . . ,xj−1,xj +1,xj+1, . . . ,xn) and polynomial difference operator of the form P(δ) = ω∈Ω ∗t.neckrasova@gmail.com Siberian Federal University. <...> All rights reserved c – 184 – cωδω, Tatiana I.Nekrasova On the Cauchy Problem for Multidimensional Difference Equations in Rational Cone where Ω ⊂ Zn is a finite set of points of n-dimensional integer lattice Zn, δω = δω1 cω are constant coefficients of the difference operator. <...> Let us consider <...>