# 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. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of confluent hypergeometric series 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 that the coefficients of the asymptotic expansion of a confluent hypergeometric series with rational parameters are in H. Finally, we prove a similar result for G-functions.
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Cited literature [29 references]

https://hal.archives-ouvertes.fr/hal-02314834
Contributor : Tanguy Rivoal <>
Submitted on : Wednesday, November 6, 2019 - 12:00:41 AM
Last modification on : Sunday, November 10, 2019 - 1:05:16 AM

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### Identifiers

• HAL Id : hal-02314834, version 2
• ARXIV : 1910.06817

### Citation

S Fischler, T. Rivoal. On Siegel's problem for E-functions. 2019. ⟨hal-02314834v2⟩

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