# Euler's factorial series and global relations

## TL;DR

This paper investigates the properties of Euler's factorial series using Padé approximations, focusing on its behavior and relations in both p-adic and Archimedean contexts.

## Contribution

It introduces new methods to analyze Euler's factorial series through Padé approximations, connecting arithmetic and analytical aspects in different valuations.

## Key findings

- Derived new relations for Euler's factorial series
- Analyzed its properties in p-adic and Archimedean valuations
- Provided insights into its arithmetic and analytical behavior

## Abstract

Using Pad\'e approximations to the series $E(z)=\sum_{k=0}^\infty k!(-z)^k$, we address arithmetic and analytical questions related to its values in both $p$-adic and Archimedean valuations.