Mathematics & Physics 2015, 8(1), 86–93 УДК 517.55+517.96 The Euler-Maclaurin Formula and Differential Operators of Infinite Order Olga A. Shishkina∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 11.09.2014, received in revised form 02.10.2014, accepted 27.11.2014 We use methods of the theory of differential operators of infinite order for solving difference equations and for generalizing the Euler-Maclaurin formula in the case of multiple summation. <...> Keywords: indefinite summation, difference equations, differential operators of infinite order. <...> Introduction The problem of summation of functions i.e the computation of the sum S (x) = x a variable upper limit x for a given function ϕ(t) is a classical one. <...> The sum of the sequence of powers of natural numbers ϕ(t) = t2 was first computed by Jakob Bernoulli. <...> His studies led to the development of several branches of the combinatorial analysis. <...> Euler proposed a method, which reduces the problem to solving the difference equation f (x+1)−f (x) = ϕ(x) , where f (x) is an unknown function, and showed that f (x) satisfies the differential equation Df (x) = ∞ µ=0 Bµ µ! <...> If the functional series on the right-hand side of the equation can be integrated term by term, then from (2) we obtain the Euler-Maclaurin formula for the solution f to equation (1), which expresses the unknown function in terms of the integral and derivatives of ϕ(t) ([1]): f (x) = ϕ(x) dx+ ∞ µ=1 Bµ µ! <...> Other approaches for solving the problem of summation we can find in [2]. <...> Recently, there has been a surge of interest in problems of summation thanks to the development of symbolic algorithms of summation of rational functions in papes by S. A.Abramov [3] and S.P.Polyakov [4], who call these problems "the indefinite summation". <...> In some cases, however, it is more appropriate to use a more general difference equation than (1). ∗olga_a_sh@mail.ru Siberian Federal University. <...> All rights reserved c – 86 – Olga A.Shishkina The Euler-Maclaurin formula and differential operators of infinite order P (δ) = p Denote δ a linear shift operator δf (x) = f (x+1) and define a polynomial <...>