  Siegel defined in 1929 two classes of power series, the E-functions and
G-functions, which generalize the Diophantine properties of the exponential and
logarithmic functions respectively. In 1949, he asked whether any E-function
can be represented as a polynomial with algebraic coefficients in a finite
number of hypergeometric E-functions with rational parameters. The case of
E-functions of differential order less than 2 was settled in the affirmative by
Gorelov in 2004, but Siegel's question is open for higher order. We prove here
that if Siegel's question has a positive answer, then the ring G of values
taken by analytic continuations of G-functions at algebraic points must be a
subring of the relatively "small" ring H generated by algebraic numbers,
$1/\pi$ and the values of the derivatives of the Gamma function at rational
points. Because that inclusion seems unlikely (and contradicts standard
conjectures), this points towards a negative answer to Siegel's question in
general. As intermediate steps, we first prove that any element of G is a
coefficient of the asymptotic expansion of a suitable E-function, which
completes previous results of ours. We then prove (in two steps) that the
coefficients of the asymptotic expansion of an hypergeometric E-function with
rational parameters are in H. Finally, we prove a similar result for
G-functions.
