Multisummability for some classes of difference equations

Braaksma, Boele L. J.; Faber, Bernard F.

doi:10.5802/aif.1511

Braaksma, Boele L. J. ; Faber, Bernard F.

Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 183-217.

Résumé
Abstract

On considère des équations aux différences finies $y (x + 1) = G (x, y)$ où $G$ prend ses valeurs dans $C^{n}$ et $G$ est méromorphe en $x$ au voisinage de $\infty$ dans $C$ et holomorphe en $y$ au voisinage de 0 dans $C^{n}$ . On montre que sous certaines conditions sur la partie linéaire de $G$ , des solutions formelles dans $C^{n} [[x^{- 1 / p}]], p \in N,$ sont multisommables. De plus on montre que ces solutions formelles se relèvent toujours en des solutions holomorphes dans des demi-plans supérieurs et inférieurs, mais généralement ces solutions ne sont pas déterminées uniquement par les solutions formelles.

This paper concerns difference equations $y (x + 1) = G (x, y)$ where $G$ takes values in $C^{n}$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $C$ and holomorphic in a neighborhood of 0 in $C^{n}$ . It is shown that under certain conditions on the linear part of $G$ , formal power series solutions in $x^{- 1 / p}, p \in N,$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1511

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     author = {Braaksma, Boele L. J. and Faber, Bernard F.},
     title = {Multisummability for some classes of difference equations},
     journal = {Annales de l'Institut Fourier},
     pages = {183--217},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     doi = {10.5802/aif.1511},
     mrnumber = {97e:39002},
     zbl = {0837.39001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1511/}
}

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Braaksma, Boele L. J.; Faber, Bernard F. Multisummability for some classes of difference equations. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 183-217. doi : 10.5802/aif.1511. http://archive.numdam.org/articles/10.5802/aif.1511/

Bibliographie
Cité par

[Bal94] W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582. Springer Verlag, Heidelberg, 1994. | MR | Zbl

[BIS] B.L.J. Braaksma, G.K. Immink, and Y. Sibuya, Cauchy-Heine and Borel transforms and Stokes phenomena, in preparation.

[Bra91] B.L.J. Braaksma, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Diff. Eq., 92 (1991), 45-75. | MR | Zbl

[Bra92] B.L.J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, 42-3 (1992), 517-540. | Numdam | MR | Zbl

[BT93] W. Balser and A. Tovbis, Multisummability of iterated integrals, Asympt. Anal., 7 (1993), 121-127. | MR | Zbl

[Duv83] A. Duval, Lemmes d'Hensel et factorisation formelle pour les opérateurs aux différences, Funkcial. Ekvac., 26 (1983), 349-368. | MR | Zbl

[Eca87] J. Ecalle, L'accélération des fonctions résurgentes, manuscrit, 1987.

[Eca93] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris, 1993.

[GLS] R. Gérard, D.A. Lutz, and R. Schäfke, Analytic reduction of some nonlinear difference equations, To appear in Funkcial. Ekvac. | Zbl

[Imm] G.K. Immink, On the summability of the formal solutions of a class of inhomogeneous linear difference equations, submitted to Funkcial. Ekvac. | Zbl

[Imm84] G.K. Immink, Asymptotics of Analytic Difference Equations, Lecture Notes in Mathematics 1085, Springer Verlag, Heidelberg, 1984. | MR | Zbl

[Imm91] G.K. Immink, Reduction to canonical forms and the Stokes phenomenon in the theory of linear difference equations, SIAM J. Math. Anal., 22 (1991), 238-259. | MR | Zbl

[Mal] B. Malgrange, Sommation des séries divergentes, Prépublication de l'Inst. Fourier, no. 268, 1994.

[Mal91] B. Malgrange, Equations différentielles à coefficients polynomiaux, Progress in Math. 96, Birkhäuser, Basel, 1991. | MR | Zbl

[MalR91] B. Malgrange and J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier, 42-1/2 (1992), 353-368. | Numdam | MR | Zbl

[MarR91] J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability, Ann. Inst. H. Poincaré, Phys. Théor., 54 (1991), 331-401. | Numdam | MR | Zbl

[Pra83] C. Praagman, The formal classification of linear difference operators, Proc. Kon. Ned. Ac. Wet., Ser A, 86 (1983), 249-261. | MR | Zbl

[Ram93] J.-P. Ramis, Séries divergentes et théories asymptotiques, Panoramas et synthèses 121, Soc. Math. France, 1993. | MR | Zbl

[vd PS] M. Van Der Put and M.F. Singer, Galois theory of difference equations, Book to appear, 1996. | Zbl

[Tur60] H.L. Turrittin, The formal theory of systems of irregular homogeneous linear difference and differential equations, Bol. Soc. Mat. Mexicana, (1960), 225-264. | MR | Zbl

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