A non-hypergeometric $E$-function

Javier Fresán; Peter Jossen

doi:https://doi.org/10.4007/annals.2021.194.3.7

Abstract

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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A non-hypergeometric $E$-function

Abstract

Authors