  Summation of the $p$-adic functional series $\sum \varepsilon^n \, n! \,
P_k^\varepsilon (n; x)\, x^n ,$ where $P_k^\varepsilon (n; x)$ is a polynomial
in $x$ and $n$ with rational coefficients, and $\varepsilon = \pm 1$, is
considered. The series is convergent in the domain $|x|_p \leq 1$ for all
primes $p$. It is found the general form of polynomials $P_k^\varepsilon (n;
x)$ which provide rational sums when $x \in \mathbb{Z}$. A class of generating
polynomials $A_k^\varepsilon (n; x)$ plays a central role in the summation
procedure. These generating polynomials are related to many sequences of
integers. This is a brief review with some new results.
