On Siegel's problem for E-functions

TL;DR

This paper investigates Siegel's problem on E-functions, showing that a positive answer would imply unlikely restrictions on the values of G-functions, thus suggesting a negative answer is more probable.

Contribution

It proves that if Siegel's question is positive, then G-function values are contained in a small algebraic ring, indicating the question is likely negative.

Findings

Any G-function value is a coefficient in an E-function's asymptotic expansion.

Coefficients of hypergeometric E-functions with rational parameters lie in a small algebraic ring.

Coefficients of G-functions' asymptotic expansions are also in this ring.

Abstract

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.