Multidimensional analogues of the Euler-Maclaurin summation formula and the Borel transform of power series

The aim of the paper is to study the problem of summation of functions of a discrete variable on integer points in a rational parallelepiped. Our method is based on Borel’s transform of power series. Integral representation for discrete antiderivative and a new variant of the Euler-Maclaurin formula are described. Consequently new identities satisfied by Bernoulli’s polynomials are obtained.