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

Abstract

The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum ∑ n−1 k=0 f (k) of values of a function f by a linear combination of a corresponding integral of f and values of its higher-order derivatives f ( j) . An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of f . It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms of the execution time and memory use. In the present paper, a multiple-sum/multi-index-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.