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Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems
Pascal Remy
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, p. 645-685

In this article, we consider a linear meromorphic differential system with several levels. We prove that the Borel transforms of its highest level’s reduced formal solutions are summable-resurgent and we give the general form of all their singularities. This one is then precised in restriction to some convenient hypotheses on the geometric configuration of singular points. Next, under the same hypotheses, we state exact formulæ to express some highest level’s Stokes multipliers of the initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already displayed by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one example.

Dans cet article, nous considérons un système différentiel linéaire méromorphe à multiples niveaux. Nous démontrons que les transformées de Borel de ses solutions formelles réduites de plus haut niveau sont résurgentes-sommables et nous donnons la forme générale de toutes leurs singularités. Celle-ci est ensuite précisée pour certaines configurations géométriques des points singuliers. Pour ces mêmes configurations, nous énonçons également des formules exactes permettant d’exprimer les multiplicateurs de Stokes de plus haut niveau du système initial à l’aide de constantes de connexion dans le plan de Borel, généralisant ainsi les formules déjà données par M. Loday-Richaud et l’auteur pour les systèmes de niveau unique. Ces formules sont illustrées par un exemple.

Received : 2015-09-06
Accepted : 2016-06-29
Published online : 2017-06-13
DOI : https://doi.org/10.5802/afst.1548
Classification:  34M03,  34M25,  34M30,  34M35,  34M40
Keywords: Linear differential system, multisummability, Stokes phenomenon, Stokes multipliers, resurgence, singularities, connection constants
@article{AFST_2017_6_26_3_645_0,
     author = {Pascal Remy},
     title = {Resurgence and highest level's connection-to-Stokes formul\ae\ for some linear meromorphic differential systems},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {3},
     year = {2017},
     pages = {645-685},
     doi = {10.5802/afst.1548},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_3_645_0}
}
Remy, Pascal. Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, pp. 645-685. doi : 10.5802/afst.1548. afst.centre-mersenne.org/item/AFST_2017_6_26_3_645_0/

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