Resurgence of the Euler-MacLaurin summation formula
Annales de l'Institut Fourier, Volume 58 (2008) no. 3, p. 893-914

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 formule sommatoire d’Euler-MacLaurin exprime la somme d’une fonction sur un réseau de points d’un intervalle comme l’addition de l’intégrale correspondante et d’un reste. Dans les cas typiques, ce reste est donné par une série asymptotique divergente du type Gevrey-1. Sous des hypothèses adéquates de décroissance de la fonction dans le demi-plan supérieur ou sur une bande verticale contenant l’intervalle de sommation, Hardy et Abel-Plana ont prouvé que cette série asymptotique est Borel sommable. Supposant que la fonction à resommer est résurgente, notre théorème principal fournit une expression, pour le reste, à la fois sous forme d’une trans-série Borel sommable et, à la fois, sous forme d’une transformée de Laplace dont l’intégrand est explicite et lui-même donné par une fonction résurgente. Notre résultat s’applique au problème d’existence de solutions d’équations différentielles linéaires avec petit paramètre, ainsi qu’à celui de la résurgence des sommes unidimensionelles de factorielles quantiques associées à des objets noués en dimension 3.

DOI : https://doi.org/10.5802/aif.2373
Classification:  34M30,  34M40
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
@article{AIF_2008__58_3_893_0,
     author = {Costin, Ovidiu and Garoufalidis, Stavros},
     title = {Resurgence of the Euler-MacLaurin summation formula},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {3},
     year = {2008},
     pages = {893-914},
     doi = {10.5802/aif.2373},
     mrnumber = {2427514},
     zbl = {1166.34055},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_3_893_0}
}
Costin, Ovidiu; Garoufalidis, Stavros. Resurgence of the Euler-MacLaurin summation formula. Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 893-914. doi : 10.5802/aif.2373. http://www.numdam.org/item/AIF_2008__58_3_893_0/

[1] Borel, E.. Sur les singularités des séries de Taylor, Bull. Soc. Math. France, Tome 26 (1898), pp. 238-248 | Numdam | MR 1504324

[2] Braaksma, B.L.J. Transseries for a class of nonlinear difference equations, J. Differ. Equations Appl., Tome 7 (2001), pp. 717-750 | Article | MR 1871576 | Zbl 1001.39002

[3] Braaksma, B.L.J.; Kuik, R. Resurgence relations for classes of differential and difference equations, Ann. Fac. Sci. Toulouse Math., Tome 13 (2004), pp. 479-492 | Article | Numdam | MR 2116813 | Zbl 1078.34068

[4] Candelpergher, B.; Nosmas, J. C.; Pham, F. Approche de la résurgence, Hermann, Actualités Mathématiques (1993) | MR 1250603 | Zbl 0791.32001

[5] Candelpergher, B.; Nosmas, J. C.; Pham, F. Premiers pas en calcul étranger, Ann. Inst. Fourier (Grenoble), Tome 43 (1993), pp. 201-224 | Article | Numdam | MR 1209701 | Zbl 0785.30017

[6] Costin, O.; Costin, R. Rigorous WKB for finite-order linear recurrence relations with smooth coefficients, SIAM J. Math. Anal., Tome 27 (1996), pp. 110-134 | Article | MR 1373150 | Zbl 0861.39007

[7] Costin, O.; Costin, R. On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations, Duke Math. J., Tome 93 (1998), pp. 289-344 | Article | MR 1625999 | Zbl 0948.34068

[8] Costin, O.; Costin, R. Global reconstruction of analytic functions from local expansions, preprint (2006)

[9] Costin, O.; Garoufalidis, S. Resurgence of the Kontsevich-Zagier power series, preprint (2006) (math.GT/0609619)

[10] Costin, O.; Garoufalidis, S. Resurgence of 1-dimensional sums of sum-product type, preprint (2007) | MR 1822494

[11] Costin, O.; Garoufalidis, S. Resurgence of the fractional polylogarithms, preprint (2007) (math.CA/0701743)

[12] Delabaere, E. Introduction to the Écalle theory, Computer algebra and differential equations, London Math. Soc. Lecture Note Series, Tome 193 (1994), pp. 59-101 | MR 1278057 | Zbl 0805.40007

[13] Delabaere, E.; Pham, F. Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor., Tome 71 (1999), pp. 1-94 | Numdam | MR 1704654 | Zbl 0977.34053

[14] Écalle, J. Resurgent functions, Vol. I–II, Mathematical Publications of Orsay, 81 (1981) | MR 670418

[15] Écalle, J. Weighted products and parametric resurgence, Analyse algébrique des perturbations singulières, I (Marseille-Luminy) Travaux en Cours, Tome 47 (1991), pp. 7-49 | MR 1296470 | Zbl 0834.34067

[16] Garoufalidis, S.; Geronimo, J. Asymptotics of q-difference equations, Contemporary Math. AMS, Tome 416 (2006), pp. 83-114 | MR 2276137 | Zbl pre05238392

[17] Garoufalidis, S.; Geronimo, J.; Le, T. T. Q. Gevrey series in quantum topology, J. Reine Angew. Math. (in press) | Zbl 1151.57006

[18] Hardy, G. H. Divergent Series, Clarendon Press, Oxford (1949) | MR 30620 | Zbl 0032.05801

[19] Jungen, R. Sur les séries de Taylor n’ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence, Comment. Math. Helv., Tome 3 (1931), pp. 266-306 | Article | Zbl 0003.11901

[20] Malgrange, B. Introduction aux travaux de J. Écalle, Enseign. Math., Tome 31 (1985), pp. 261-282 | MR 819354 | Zbl 0601.58043

[21] Olver, F. W. J. Asymptotics and special functions, A K Peters, Ltd., Wellesley, MA, Reprint. AKP Classics (1997) | MR 1429619 | Zbl 0982.41018

[22] Sauzin, D. Resurgent functions and splitting problems, preprint (2006)

[23] Zagier, D. Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, Tome 40 (2001), pp. 945-960 | Article | MR 1860536 | Zbl 0989.57009