Summation of $p$-Adic Functional Series in Integer Points

Branko Dragovich, Andrei Yu. Khrennikov, Nataša Z. Mišić

Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all $p$-adic cases. The corresponding infinite functional series are divergent in the real case, but they are convergent and have $p$-adic invariant sums in $p$-adic cases. We found polynomials which generate all significant ingredients of these series and make connection between their real and $p$-adic properties. In particular, we found connection of one of our integer sequences with the Bell numbers.