Abstract:
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 primes, 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$. The paper formulates conjectures, related to the values of the considered series.
Keywords:transcendence,summatuon of a series, polyadic number.
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