  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 Andr\'e's theory of E-operators and Katz's computation of
the Galois group of hypergeometric differential equations.
