Mathematics & Physics 2015, 8(4), 437–441 УДК 517.55 On the Correctness of Polynomial Difference Operators Marina S. Rogozina∗ Krasnoyarsk State Medical University Partizana Zheleznyaka, 1, Krasnoyarsk, 660022 Russia Received 10.09.2015, received in revised form 17.10.2015, accepted 01.11.2015 The correctness Cauchy problem is explored for a polynomial difference operator. <...> The easily verifiable sufficient condition correctness for the Cauchy problem for a polynomial difference operator with constant coefficients is proved whose characteristic polynomial is homogeneous. <...> Together with the method of generating functions, they yield a powerful machinery for studying enumeration problems in combinatorial analysis (see, e.g. [1]). <...> Another source of difference equations is the discretization of differential equations. <...> For instance, the discretization of Cauchy-Riemann equations led to the appearance of the theory of discrete analytic functions (see, e.g. [2, 3]) which found applications in the theory of Reimannian surfaces and combinatorial analysis (see, e.g. [4, 5]). <...> The methods of discretization of a differential problem comprise an important part of the theory of difference schemes, and they also lead to difference equations (see, e.g. [6]). <...> The theory of difference schemes is exploring ways of constructing difference schemes, explores the challenges posed difference and the convergence of the solution of the difference problem to the solution of the original differential problem, engaged justification of algorithms for solving problems of difference. <...> Let us select a subset X0 ⊂ X of the «initial» («boundary») points from the set X. Let us formulate the problem. <...> Find a function f(x) that satisfies the equation (1) and coincides with the given function on set X0 f(x) = ϕ(x), x ∈ X0. ∗rogozina.marina@mail.ru ⃝ Siberian Federal University. <...> All rights reserved c – 437 – (2) Marina S. Rogozina On the Correctness of Polynomial Difference Operators The problem (1)–(2) is called the Cauchy problem for a polynomial difference operatorP (δ). <...> It is said (see, e.g. [7]) that the problem of the form (1)–(2) for a polynomial difference operator P (δ) posed, if the following conditions are satisfied: a) the problem is uniquely solvable for any initial data ϕ(x) and right-hand sides g(x); b) there are constants M1 > 0 <...>