Summation of p-Adic Functional Series in Integer Points

TL;DR

This paper explores the summation of factorial-containing functional series, revealing their divergence in real numbers but convergence in p-adic numbers, and establishes connections with Bell numbers and polynomial generators.

Contribution

It introduces a method to analyze p-adic convergence of factorial series and links these to polynomial structures and Bell numbers, advancing understanding of p-adic series summation.

Findings

Finite sums are identical in real and p-adic cases.

Series diverge in real numbers but converge p-adically.

Connection established between integer sequences and Bell numbers.

Abstract

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.