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.
Javier Fresán. Peter Jossen. "A non-hypergeometric E-function." Ann. of Math. (2) 194 (3) 903 - 942, November 2021. https://doi.org/10.4007/annals.2021.194.3.7