Multisummability for some classes of difference equations. (English) Zbl 0837.39001

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

Ann. Inst. Fourier 46, No. 1, 183-217 (1996).

Summary: This paper concerns difference equations \(y(x+1)=G(x,y)\) where \(G\) takes values in \({\mathbb{C}}^n\) and \(G\) is meromorphic in \(x\) in a neighborhood of \(\infty\) in \({\mathbb{C}}\) and holomorphic in a neighborhood of 0 in \({\mathbb{C}}^n\). It is shown that under certain conditions on the linear part of \(G\), formal power series solutions in \(x^{-1/p}, p\in {\mathbb{N}},\) 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.

MSC:

39A10	Additive difference equations
40G10	Abel, Borel and power series methods
32A10	Holomorphic functions of several complex variables

Keywords:

multisummability; normal forms; Borel and Laplace transforms; Gevrey series; Stokes phenomenon; difference equations; formal power series solutions; holomorphic solutions

References:

[1]	[Bal94] , From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582. Springer Verlag, Heidelberg, 1994. · Zbl 0810.34046
[2]	[BIS] , , and , Cauchy-Heine and Borel transforms and Stokes phenomena, in preparation.
[3]	[Bra91] , Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Diff. Eq., 92 (1991), 45-75. · Zbl 0729.34005
[4]	[Bra92] , Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, 42-3 (1992), 517-540. · Zbl 0759.34003
[5]	[BT93] and , Multisummability of iterated integrals, Asympt. Anal., 7 (1993), 121-127. · Zbl 0787.30026
[6]	[Duv83] , Lemmes d’Hensel et factorisation formelle pour les opérateurs aux différences, Funkcial. Ekvac., 26 (1983), 349-368. · Zbl 0543.12018
[7]	[Eca87] , L’accélération des fonctions résurgentes, manuscrit, 1987.
[8]	[Eca93] , Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris, 1993. · Zbl 1241.34003
[9]	[GLS] , , and , Analytic reduction of some nonlinear difference equations, To appear in Funkcial. Ekvac. · Zbl 0847.39003
[10]	[Imm] , On the summability of the formal solutions of a class of inhomogeneous linear difference equations, submitted to Funkcial. Ekvac. · Zbl 0872.39002
[11]	[Imm84] , Asymptotics of Analytic Difference Equations, Lecture Notes in Mathematics 1085, Springer Verlag, Heidelberg, 1984. · Zbl 0548.39001
[12]	[Imm91] , Reduction to canonical forms and the Stokes phenomenon in the theory of linear difference equations, SIAM J. Math. Anal., 22 (1991), 238-259. · Zbl 0733.39004
[13]	[Mal] , Sommation des séries divergentes, Prépublication de l’Inst. Fourier, no. 268, 1994.
[14]	[Mal91] , Equations différentielles à coefficients polynomiaux, Progress in Math. 96, Birkhäuser, Basel, 1991. · Zbl 0764.32001
[15]	[MalR91] and , Fonctions multisommables, Ann. Inst. Fourier, 42-1/2 (1992), 353-368. · Zbl 0759.34007
[16]	[MarR91] and , Elementary acceleration and multisummability, Ann. Inst. H. Poincaré, Phys. Théor., 54 (1991), 331-401. · Zbl 0748.12005
[17]	[Pra83] , The formal classification of linear difference operators, Proc. Kon. Ned. Ac. Wet., Ser A, 86 (1983), 249-261. · Zbl 0519.39003
[18]	[Ram93] , Séries divergentes et théories asymptotiques, Panoramas et synthèses 121, Soc. Math. France, 1993. · Zbl 0830.34045
[19]	[vd PS] and , Galois theory of difference equations, Book to appear, 1996. · Zbl 0930.12006
[20]	[Tur60] , The formal theory of systems of irregular homogeneous linear difference and differential equations, Bol. Soc. Mat. Mexicana, (1960), 225-264. · Zbl 0100.08201