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

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Department of Mathematical Sciences


The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum ∑n−1k=0f(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.

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arXiv:1705.09159 [math.CA]