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Irregular modified $ A$-hypergeometric systems


Authors: Francisco-Jesús Castro-Jiménez, María-Cruz Fernández-Fernández, Tatsuya Koike and Nobuki Takayama
Journal: Trans. Amer. Math. Soc. 367 (2015), 5415-5445
MSC (2010): Primary 32C38, 14F10; Secondary 33C70, 13N10, 14M25
DOI: https://doi.org/10.1090/S0002-9947-2014-06225-9
Published electronically: November 24, 2014
MathSciNet review: 3347178
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Abstract: A modified $ A$-hypergeometric system is a system of differential equations for the function $ f(t^w \cdot x)$, where $ f(y)$ is a solution of an $ A$-hypergeometric system in $ n$ variables and $ w$ is an $ n$-dimensional integer vector, which is called the weight vector. We study the irregularity of modified systems by adapting to this case the notion of umbrella introduced by M. Schulze and U. Walther. Especially, we study slopes and Gevrey series solutions. We develop some applications of this study. Under some conditions we give Laplace integral representations of divergent series solutions of the modified system and we show that certain Gevrey series solutions of the original $ A$-hypergeometric system along coordinate varieties are Gevrey asymptotic expansions of holomorphic solutions of the $ A$-hypergeometric system.


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Additional Information

Francisco-Jesús Castro-Jiménez
Affiliation: Department of Algebra, University of Sevilla, Seville, Spain
Email: castro@us.es

María-Cruz Fernández-Fernández
Affiliation: Department of Algebra, University of Sevilla, Seville, Spain
Email: mcferfer@us.es

Tatsuya Koike
Affiliation: Department of Mathematics, University of Kobe, Kobe, Japan
Email: koike@math.kobe-u.ac.jp

Nobuki Takayama
Affiliation: Department of Mathematics, University of Kobe, Kobe, Japan
Email: takayama@math.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2014-06225-9
Received by editor(s): September 9, 2012
Received by editor(s) in revised form: May 28, 2013
Published electronically: November 24, 2014
Additional Notes: The first and second authors were partially supported by MTM2010-19336 and FEDER and Junta de Andalucía FQM5849, FQM333
The first author was also partially supported by S-13025-JSPS (Japan)
The second author was also partially supported by Max Planck Institute für Mathematik (Bonn)
The third author was partially supported by JSPS grants-in-aid No. 21740098 and No. S-24224001.
Article copyright: © Copyright 2014 American Mathematical Society