p-Adic Invariant Summation of Some p-Adic Functional Series

TL;DR

This paper investigates p-adic series involving factorials that converge for all primes and share a common integer sum, introducing polynomial sequences that generate related number-theoretic and combinatorial sequences.

Contribution

It presents a large class of p-adic factorial series with integer values, and constructs polynomial sequences via recurrence relations relevant to number theory and combinatorics.

Findings

Identified p-adic series with uniform integer sums for all primes

Developed polynomial sequences generating related number-theoretic sequences

Established recurrence relations for constructing these polynomials

Abstract

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.