# On Siegel’s problem for $E$-functions

### Stéphane Fischler

CNRS, Université Paris-Saclay, Orsay, France### Tanguy Rivoal

CNRS, Université Grenoble Alpes, France

## 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. He asked whether any $E$-function can be represented as a polynomial with algebraic coefficients in a finite number of $E$-functions of the form $_{p}F_{q}(λz_{q−p+1})$, $q≥p≥1$, with rational parameters. The case of $E$-functions of differential order less than or equal to 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, $1/π$ 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 a hypergeometric $E$-function with rational parameters are in $H$. Finally, we prove a similar result for $G$-functions.

## Cite this article

Stéphane Fischler, Tanguy Rivoal, On Siegel’s problem for $E$-functions. Rend. Sem. Mat. Univ. Padova 148 (2022), pp. 83–115

DOI 10.4171/RSMUP/107