A non-hypergeometric $E$-function

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.

Authors

Javier Fresán

CMLS, École polytechnique, Palaiseau, France

Peter Jossen

ETH Zürich, Zürich, Switzerland and King's College, London, England