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Article Dans Une Revue Rendiconti del Seminario Matematico della Università di Padova Année : 2022

On Siegel's problem for E-functions

Résumé

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.
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Dates et versions

hal-02314834 , version 1 (14-10-2019)
hal-02314834 , version 2 (06-11-2019)
hal-02314834 , version 3 (29-05-2020)

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Stéphane Fischler, Tanguy Rivoal. On Siegel's problem for E-functions. Rendiconti del Seminario Matematico della Università di Padova, 2022, 148, pp.83-115. ⟨10.4171/RSMUP/107⟩. ⟨hal-02314834v3⟩
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