# zbMATH — the first resource for mathematics

Resurgence relations for classes of differential and difference equations. (English) Zbl 1078.34068
The paper is devoted to the study of meromorphic differential and difference equations of level 1. It is shown that small solutions of these equations can be represented as exponential series with Borel sums of certain formal series as coefficients. The Stokes transition in these representations is considered in the case that the sum of the coefficients is changed between sectors. Finally, resurgence relations are derived by using one-sided Borel transforms and a sketch is given of the bridge equation of Ecalle and its use for the above-mentioned equations.

##### MSC:
 34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain 34M37 Resurgence phenomena (MSC2000) 39A10 Additive difference equations
Full Text:
##### References:
 [1] Braaksma ( B. ) , Kuik ( R. ). - Asymptotics and singularities for a class of difference equation , Contemporary Mathematics , to appear ( 2004 ). MR 2130828 | Zbl 1076.39005 · Zbl 1076.39005 [2] Braaksma ( B.L.J. ). - Transseries for a class of nonlinear difference equations , J. Diff. Equations Appl. , 7 ( 5 ), p. 717 - 750 ( 2001 ). MR 1871576 | Zbl 1001.39002 · Zbl 1001.39002 · doi:10.1080/10236190108808298 [3] Candelpergher ( B. ) , Nosmas ( J.-C. ) , Pham ( F. ). - Approche de la résurgence , Actualités Mathématiques , Herman , Paris ( 1993 ). MR 1250603 | Zbl 0791.32001 · Zbl 0791.32001 [4] Costin ( O. ). - On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations , Duke Math. J. , 93 ( 2 ), p. 289 - 344 ( 1998 ). Article | MR 1625999 | Zbl 0948.34068 · Zbl 0948.34068 · doi:10.1215/S0012-7094-98-09311-5 · minidml.mathdoc.fr [5] Ecalle ( J. ). - Les fonctions résurgentes . Tome III , Université de Paris-Sud Département de Mathématique , Orsay ( 1985 ). Article | MR 852210 | Zbl 0602.30029 · Zbl 0602.30029 · minidml.mathdoc.fr [6] Ecale ( J. ). - Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac , Actualités Mathématiques , Hermann , Paris ( 1992 ). MR 1399559 [7] Kuik ( R. ) Transseries in difference and differential equations , Thesis, University of Groningen ( 2003 ). [8] Malgrange ( B. ). - Introduction aux travaux de J. Ecalle , Enseign. Math. ( 2 ) 31 ( 3 - 4 ), p. 261 - 282 ( 1985 ). MR 819354 | Zbl 0601.58043 · Zbl 0601.58043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.