Resurgence of the Euler-Maclaurin summation formula
type: Item
instance of: scholarly article
title: Resurgence of the Euler-Maclaurin summation formula
zbMATH Open document ID: 1166.34055
author: O. Costin
author: Stavros Garoufalidis
publication date: 2008-07-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0703641
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10338
review text: The Euler-MacLaurin summation formula relates summation to integration as follows: \[ \sum _{k=1}^Nf(k/N)=N\int _0^1f(s)ds+(1/2)(f(1)-f(0))+R(f,N) \] where the remainder term \(R(f,N)\) has an asymptotic expansion given in terms of Bernoulli numbers and values of the derivatives of \(f\) at \(0\) and \(1\). For a typical analytic function this expansion is a divergent Gevrey-1 series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) proved that the asymptotic expansion is a Borel summable series, and gave an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, the authors give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, their summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. Finally, they give two applications of the results. One concerns the construction of solutions of linear difference equations with a small parameter. Another concerns resurgence of 1-dimensional sums of quantum factorials, that are associated to knotted 3-dimensional objects.
Mathematics Subject Classification ID: 34M37
Mathematics Subject Classification ID: 34M30
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 5298325
zbMATH Keywords: Euler-Maclaurin summation formula
zbMATH Keywords: Abel-Plana formula
zbMATH Keywords: Stirling's formula
zbMATH Keywords: resurgence
zbMATH Keywords: resurgent functions
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Borel transform
zbMATH Keywords: Borel summation
zbMATH Keywords: Laplace transform
zbMATH Keywords: transseries
zbMATH Keywords: parametric resurgence
zbMATH Keywords: co-equational resurgence
zbMATH Keywords: WKB
zbMATH Keywords: difference equations with a parameter
zbMATH Keywords: quantum topology
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
OpenAlex ID: W2114625147
arXiv ID: math/0703641
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DOI: 10.5802/AIF.2373
published in: Annales de l’institut Fourier
description: scientific article