Resurgence of the Euler-MacLaurin summation formula

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

top

Abstract

top
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

top

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 -

References

top
1. E.. Borel, Sur les singularités des séries de Taylor, Bull. Soc. Math. France 26 (1898), 238-248 Zbl29.0210.03MR1504324
2. B.L.J. Braaksma, Transseries for a class of nonlinear difference equations, J. Differ. Equations Appl. 7 (2001), 717-750 Zbl1001.39002MR1871576
3. B.L.J. Braaksma, R. Kuik, Resurgence relations for classes of differential and difference equations, Ann. Fac. Sci. Toulouse Math. 13 (2004), 479-492 Zbl1078.34068MR2116813
4. B. Candelpergher, J. C. Nosmas, F. Pham, Approche de la résurgence, (1993), Hermann Zbl0791.32001MR1250603
5. B. Candelpergher, J. C. Nosmas, F. Pham, Premiers pas en calcul étranger, Ann. Inst. Fourier (Grenoble) 43 (1993), 201-224 Zbl0785.30017MR1209701
6. O. Costin, R. Costin, Rigorous WKB for finite-order linear recurrence relations with smooth coefficients, SIAM J. Math. Anal. 27 (1996), 110-134 Zbl0861.39007MR1373150
7. O. Costin, R. Costin, On Borel summation and Stokes phenomena for rank-$1$ nonlinear systems of ordinary differential equations, Duke Math. J. 93 (1998), 289-344 Zbl0948.34068MR1625999
8. O. Costin, R. Costin, Global reconstruction of analytic functions from local expansions, preprint (2006) Zbl1120.52002
9. O. Costin, S. Garoufalidis, Resurgence of the Kontsevich-Zagier power series, preprint (2006) Zbl1238.57016
10. O. Costin, S. Garoufalidis, Resurgence of 1-dimensional sums of sum-product type, preprint (2007) MR1822494
11. O. Costin, S. Garoufalidis, Resurgence of the fractional polylogarithms, preprint (2007) Zbl1201.30044
12. E. Delabaere, Introduction to the Écalle theory, Computer algebra and differential equations, London Math. Soc. Lecture Note Series 193 (1994), 59-101 Zbl0805.40007MR1278057
13. E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 1-94 Zbl0977.34053MR1704654
14. J. Écalle, Resurgent functions, Vol. I–II, (1981), Mathematical Publications of Orsay, 81 MR670418
15. J. Écalle, Weighted products and parametric resurgence, Analyse algébrique des perturbations singulières, I (Marseille-Luminy) Travaux en Cours 47 (1991), 7-49 Zbl0834.34067MR1296470
16. S. Garoufalidis, J. Geronimo, Asymptotics of $q$-difference equations, Contemporary Math. AMS 416 (2006), 83-114 Zbl1144.57005MR2276137
17. S. Garoufalidis, J. Geronimo, T. T. Q. Le, Gevrey series in quantum topology, J. Reine Angew. Math. Zbl1151.57006
18. G. H. Hardy, Divergent Series, (1949), Clarendon Press, Oxford Zbl0032.05801MR30620
19. R. Jungen, Sur les séries de Taylor n’ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence, Comment. Math. Helv. 3 (1931), 266-306 Zbl0003.11901
20. B. Malgrange, Introduction aux travaux de J. Écalle, Enseign. Math. 31 (1985), 261-282 Zbl0601.58043MR819354
21. F. W. J. Olver, Asymptotics and special functions, (1997), A K Peters, Ltd., Wellesley, MA Zbl0982.41018MR1429619
22. D. Sauzin, Resurgent functions and splitting problems, preprint (2006)
23. D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 (2001), 945-960 Zbl0989.57009MR1860536

top

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.