On Siegel’s problem for E-functions
Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 83-115.
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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 ), qp1, 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.

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DOI : 10.4171/rsmup/107
Classification : 33, 11, 41
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     title = {On {Siegel{\textquoteright}s} problem for $E$-functions},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {83--115},
     volume = {148},
     year = {2022},
     doi = {10.4171/rsmup/107},
     mrnumber = {4542374},
     zbl = {1512.33011},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/107/}
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Stéphane Fischler; Tanguy Rivoal. On Siegel’s problem for $E$-functions. Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 83-115. doi : 10.4171/rsmup/107. http://www.numdam.org/articles/10.4171/rsmup/107/

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