A non-hypergeometric E-function

We answer in the negative Siegel's question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Authors: Fresán, Javier, Jossen, Peter
Format: Article
Language:English
Subjects:
Differential equations
Fourier transforms
Functions (mathematics)
Operators (mathematics)
Polynomials
Singularity (mathematics)
Online Access:Get full text
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Summary:We answer in the negative Siegel's question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André's theory of E-operators and Katz's computation of the Galois group of hypergeometric differential equations.
ISSN:2331-8422
DOI:10.48550/arxiv.2012.11005