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

Authors : Iosif PİNELİS

Pages : 72-92

Doi:10.33205/cma.1102689

View : 22 | Download : 9

Publication Date : 2022-06-15

Article Type : Research Paper

Abstract :The Euler--Maclaurin insert ignore into journalissuearticles values(EM); summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} finsert ignore into journalissuearticles values(k);$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{insert ignore into journalissuearticles values(j);}$. An alternative insert ignore into journalissuearticles values(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/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.
Keywords : Euler Maclaurin summation formula, alternative summation formula, multiple sums, multi index series, approximation, lattice polytopes