Arithmetic properties of polyadic integers

Arithmetic properties of series of the form
$$\sum_{n=0}^\infty a_{n}\cdot n!$$
with $a_n\in\mathbb Z$ are studied.
The concept of infinite algebraic independence polyadic numbers.
A theorem on the algebraic independence polyadic infinite number of class $ F\left(\mathbb {Q}, C_1, C_2, C_3, d_ {0} \right) $, if they are connected by a system of linear differential equations of a certain kind.
Bibliography: 9 titles.