Sommation effective d’une somme de Borel par séries de factorielles

Delabaere, Eric; Rasoamanana, Jean-Marc

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 421-456.

Nous abordons dans cet article la question de la sommation effective d’une somme de Borel d’une série par la série de factorielles associée. Notre approche fournit un contrôle de l’erreur entre la somme de Borel recherchée et les sommes partielles de la série de factorielles. Nous généralisons ensuite cette méthode au cadre des séries de puissances fractionnaires, après avoir démontré un analogue d’un théorème de Nevanlinna de sommation de Borel fine pour ce cadre.

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

[1] Balser, W.; Lutz, D. A.; Schäfke, R. On the convergence of Borel approximants, J. Dynam. Control Systems, Volume 8 (2002) no. 1, pp. 65-92 | DOI | MR | Zbl

[2] Berry, M. V.; Howls, C. J. Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London Ser. A, Volume 434 (1991) no. 1892, pp. 657-675 | DOI | MR | Zbl

[3] Canalis-Durand, M. Solutions Gevrey d’équations différentielles singulièrement perturbées (1999) (Thèse d’habilitation à diriger des recherches)

[4] Candelpergher, B.; Nosmas, J.-C.; Pham, F. Approche de la résurgence, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1993 | MR | Zbl

[5] Comtet, Louis Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, 1974 (The art of finite and infinite expansions) | MR | Zbl

[6] Delabaere, E. Effective resummation methods for an implicit resurgent function (2006) (submitted)

[7] Delabaere, E.; Howls, C. J. Global asymptotics for multiple integrals with boundaries, Duke Math. J., Volume 112 (2002) no. 2, pp. 199-264 | DOI | MR | Zbl

[8] Delabaere, E.; Rasoamanana, J.-M. Resurgent deformations for an ODE of order 2, Pacific Journal of Mathematics, Volume 223 (2006) no. 1, pp. 35-93 | DOI | MR | Zbl

[9] Delabaere, Eric Introduction to the Écalle theory, Computer algebra and differential equations (1992) (London Math. Soc. Lecture Note Ser.), Volume 193, Cambridge Univ. Press, Cambridge, 1994, pp. 59-101 | MR | Zbl

[10] Delabaere, Eric; Pham, Frédéric Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor., Volume 71 (1999) no. 1, pp. 1-94 | EuDML | Numdam | MR | Zbl

[11] Dingle, R. B. Asymptotic expansions : their derivation and interpretation, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York,, 1973 | MR | Zbl

[12] Écalle, Jean Les fonctions résurgentes. Tome I, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], 5, Université de Paris-Sud Département de Mathématique, Orsay, 1981 (Les algèbres de fonctions résurgentes. [The algebras of resurgent functions], With an English foreword) | MR | Zbl

[13] Écalle, Jean Les fonctions résurgentes. Tome II, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], 6, Université de Paris-Sud Département de Mathématique, Orsay, 1981 (Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration]) | MR | Zbl

[14] Écalle, Jean Les fonctions résurgentes. Tome III, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (L’équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects]) | MR | Zbl

[15] Feller, William An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons Inc., New York, 1968 | MR | Zbl

[16] Jentschura, U. Quantum Electrodynamic, Bound-State Calculations and Large-Order Perturbation Theory (2004) (Habilitation Thesis, Dresden University of Technology, 3rd edition)

[17] Malgrange, Bernard Sommation des séries divergentes, Exposition. Math., Volume 13 (1995) no. 2-3, pp. 163-222 | MR | Zbl

[18] Nevanlinna, F. Zur Theorie der Asymptotischen Potenzreihen, Suomalaisen Tiedeakatemian Kustantama, Helsinki, 1918

[19] Nörlund, N.E. Leçons sur les Séries d’Interpolation, Gautier-Villars, Paris, 1926 | JFM

[20] Olde Daalhuis, A. B. Hyperterminants. I, J. Comput. Appl. Math., Volume 76 (1996) no. 1-2, pp. 255-264 | DOI | MR | Zbl

[21] Olde Daalhuis, A. B. Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 454 (1998) no. 1968, pp. 1-29 | DOI | MR | Zbl

[22] Olde Daalhuis, A. B. Hyperterminants. II, J. Comput. Appl. Math., Volume 89 (1998) no. 1, pp. 87-95 | DOI | MR | Zbl

[23] Poincaré, H. Les méthodes nouvelles de la mécanique céleste. Tome I, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Librairie Scientifique et Technique Albert Blanchard, Paris, 1987 (Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotiques. [Periodic solutions. Nonexistence of uniform integrals. Asymptotic solutions], Reprint of the 1892 original, With a foreword by J. Kovalevsky, Bibliothèque Scientifique Albert Blanchard. [Albert Blanchard Scientific Library]) | JFM | MR | Zbl

[24] Ramis, Jean-Pierre; Schäfke, Reinhard Gevrey separation of fast and slow variables, Nonlinearity, Volume 9 (1996) no. 2, pp. 353-384 | DOI | MR | Zbl

[25] Simon, B. Large orders and summability of eigenvalue perturbation theory : a mathematical overview., International Journal of Quantum Chemistry, Volume XXI (1982), pp. 3-25 | DOI

[26] Stokes, G.G. On the Discontinuity of arbitrary constants which appear in divergent developments, Transactions of the Cambridge Philosophical Society, Volume X (1857), pp. Part I

[27] Thomann, Jean Resommation des series formelles. Solutions d’équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières, Numer. Math., Volume 58 (1990) no. 5, pp. 503-535 | DOI | EuDML | MR | Zbl

[28] Wasow, Wolfgang Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965 | MR | Zbl

[29] Watson, G.N. The transformation of an asymptotic series into a convergent series of inverse factorials, Cir. Mat. Palermo, Volume 34 (1912), pp. 41-88 | DOI | JFM