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A non-hypergeometric E-function

Research output: Contribution to journalArticlepeer-review

Javier Fresàn, Peter Jossen

Original languageEnglish
Pages (from-to)903-942
Number of pages40
Issue number3
PublishedNov 2021

Bibliographical note

Funding Information: Keywords: E-function, hypergeometric series, differential Galois theory, Fourier-Laplace transform AMS Classification: Primary: 11J91, 33C20, 34M35. The research of J. F. was partially supported by the grant ANR-18-CE40-0017 of Agence Nationale de la Recherche. © 2021 Department of Mathematics, Princeton University. Publisher Copyright: © 2021. Department of Mathematics, Princeton University

King's Authors


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

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