p-Adic invariant summation of some p-adic functional series

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A _k (n; x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.