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.