Approximating sums by integrals only: multiple sums and sums over lattice polytopes

The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum Sigma(n-1)(k=0) f(k) of values of a function f by a linear combination of a corresponding integral off and values of its higher-order derivatives f((j)). An alternative (Alt) summation formula was presented by the author, which approximates the sum by a linear combination of integrals only, without using derivatives of f. It was shown that the Alt formula will in most cases outperform the EM formula. In the present paper, a multiple-sum/multiindex-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.