Arithmetic properties of values of divergent in $\mathbb{C}$ series

V. G. Chirskii

Čebyševskij sbornik, Tome 25 (2024) no. 3, pp. 259-269

Résumé

The article describes the directions of research on the arithmetic properties of series values of the form $$\sum_{n=0}^{\infty}a_{n}\cdot n!z^{n}$$ with coefficients $a_{n}$ satisfying certain conditions. Under these conditions, the considered series, other than the polynomial, converges in the field $\mathbb{C} $ only at $z=0$. However, for almost all but a finite number of prime $p$ numbers, such a series converges in the fields $\mathbb{Q}_p.$ Therefore there are two ways of research. We can either consider the arithmetic properties of the result of some summation of this series, or consider the values of this series in the field $ \mathbb{Q}_p$. An example of the first approach is the series considered by Euler $$1-x+2x^{2}+\dots+(-1)^{n}\cdot n!x^{n}+\dots,$$ as a result of the summation of which, with the substitution $x=1$, we obtain the remarkable equality $$\sum_{n=0}^{\infty} (-1)^{n}n!= e(\gamma -\sum_{n=1}^{\infty} \frac{1}{n\cdot n!}),$$ where $\gamma$ – Euler's constant. Another direction of research uses the concept of global relation introduced by E. Bombieri. Using the modified Siegel—Shidlovskii method, it is possible to obtain analogues of the main theorems of A.B. Shidlovskii for $E-$ functions. The use of Hermite-Pade approximants made it possible to consider the values of generalized hypergeometric series not only with algebraic, but also with certain parameters transcendental in any field $ \mathbb{Q}_p$.

Détail

Keywords: transcendence,summatuon of a series, polyadic number.