On Siegel’s problem for $E$-functions

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 ${}_p{F}_q(\lambda z^{q-p+1})$, $q\ge p\ge 1$, 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 $\mathbf{G}$ of values taken by analytic continuations of $G$-functions at algebraic points must be a subring of the relatively “small” ring $\mathbf{H}$ 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 $\mathbf{G}$ 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 $\mathbf{H}$. Finally, we prove a similar result for $G$-functions.