A non-hypergeometric E-function

TL;DR

This paper demonstrates that not all E-functions are polynomial expressions of hypergeometric E-functions by analyzing the singularities of their Fourier transforms and applying advanced algebraic theories.

Contribution

It provides a negative answer to Siegel's question by establishing constraints on the Fourier transform singularities of certain E-functions.

Findings

E-functions outside the hypergeometric class exist.

Fourier transform singularities of certain E-functions are constrained.

The result relies on Andre9's E-operator theory and Galois group computations.

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