In a landmark 1929 paper, Siegel introduced the class of E-functions with the goal of
generalising the transcendence theorems for the values of the exponential. E-functions are
power series with algebraic coefficients subject to certain growth conditions of arithmetic
nature that satisfy a linear differential equation.
Besides the exponential, examples include Bessel functions and a rich family of
hypergeometric series. Siegel asked whether all E-functions are polynomial expressions in
these hypergeometric series.

I will explain why the answer is negative and a possible amendment to Siegel's question in
the form "all E-functions come from exponential motives".
This is a joint work with P. Jossen.