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Arithmetic properties of Euler series. (English. Russian original) Zbl 1330.11047

Mosc. Univ. Math. Bull. 70, No. 1, 41-43 (2015); translation from Vestn. Mosk. Univ., Ser. I 70, No. 1, 59-61 (2015).
Summary: A lower bound for the \(p\)-adic valuation of the number \(E_p= \Sigma_{n=1}^{\infty}n! \in \mathbb Q_p\) defined by an Euler-type series is proved in the paper for infinitely many prime numbers \(p\).

MSC:

11J61 Approximation in non-Archimedean valuations
11J72 Irrationality; linear independence over a field
11J85 Algebraic independence; Gel’fond’s method
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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References:

[1] V. G. Chirskii, “On Global Relations,” Matem. Zametki 48(2), 123 (1990). · Zbl 0764.11031
[2] D. Bertrand, V. Chirskii, and J. Yebbon, “Effective Estimates for Global Relations on Euler-Type Series,” Ann. Fac. Sci. Toulouse XIII(2), 241 (2004). · Zbl 1176.11036 · doi:10.5802/afst.1069
[3] Yu. V. Nesterenko, “Hermite-Pade Approximations of Generalized Hypergeometric Functions,” Matem. Sborn, 185(10), 39 (1994). · Zbl 0849.11052
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