On Summation of $p$-Adic Series

TL;DR

This paper investigates the summation of specific $p$-adic series involving factorials and polynomials, identifying conditions for rational sums and introducing generating polynomials linked to integer sequences.

Contribution

It determines the general form of polynomials that yield rational sums in $p$-adic series and introduces generating polynomials central to the summation process.

Findings

Series converges in $|x|_p \\leq 1$ for all primes $p$

Identifies polynomials $P_k^\ ext{\varepsilon}$ that produce rational sums when $x \\in \\mathbb{Z}$

Introduces generating polynomials $A_k^\ ext{\varepsilon}$ related to integer sequences

Abstract

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.