Resurgence of the Euler-MacLaurin summation formula

Ovidiu Costin[1]; Stavros Garoufalidis[2]

  • [1] Ohio State University Department of Mathematics 231 W 18th Avenue Columbus, OH 43210 (USA)
  • [2] Georgia Institute of Technology School of Mathematics Atlanta, GA 30332-0160 (USA)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 3, page 893-914
  • ISSN: 0373-0956

Abstract

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The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it 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) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula.Using a mild resurgence hypothesis for the function to be summed, we 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, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval.Finally, we give two applications of our 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.

How to cite

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Costin, Ovidiu, and Garoufalidis, Stavros. "Resurgence of the Euler-MacLaurin summation formula." Annales de l’institut Fourier 58.3 (2008): 893-914. <http://eudml.org/doc/10338>.

@article{Costin2008,
abstract = {The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it 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) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula.Using a mild resurgence hypothesis for the function to be summed, we 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, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval.Finally, we give two applications of our 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.},
affiliation = {Ohio State University Department of Mathematics 231 W 18th Avenue Columbus, OH 43210 (USA); Georgia Institute of Technology School of Mathematics Atlanta, GA 30332-0160 (USA)},
author = {Costin, Ovidiu, Garoufalidis, Stavros},
journal = {Annales de l’institut Fourier},
keywords = {Euler-MacLaurin summation formula; Abel-Plana formula; resurgence; resurgent functions; Bernoulli numbers; Borel transform; Borel summation; Laplace transform; transseries; parametric resurgence; co-equational resurgence; WKB; difference equations with a parameter; Stirling’s formula; Quantum Topology; Stirling's formula; quantum topology},
language = {eng},
number = {3},
pages = {893-914},
publisher = {Association des Annales de l’institut Fourier},
title = {Resurgence of the Euler-MacLaurin summation formula},
url = {http://eudml.org/doc/10338},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Costin, Ovidiu
AU - Garoufalidis, Stavros
TI - Resurgence of the Euler-MacLaurin summation formula
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 3
SP - 893
EP - 914
AB - The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it 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) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula.Using a mild resurgence hypothesis for the function to be summed, we 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, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval.Finally, we give two applications of our 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.
LA - eng
KW - Euler-MacLaurin summation formula; Abel-Plana formula; resurgence; resurgent functions; Bernoulli numbers; Borel transform; Borel summation; Laplace transform; transseries; parametric resurgence; co-equational resurgence; WKB; difference equations with a parameter; Stirling’s formula; Quantum Topology; Stirling's formula; quantum topology
UR - http://eudml.org/doc/10338
ER -

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