Euler's factorial series at algebraic integer points

TL;DR

This paper investigates the non-vanishing of linear forms in values of Euler's factorial series at algebraic integer points over number fields, providing new lower bounds and extending results to primes in residue classes.

Contribution

It introduces new non-vanishing results and explicit Padé approximations for generalized factorial series, advancing understanding of Euler's series at algebraic points.

Findings

Established non-vanishing results for linear forms in Euler's series values.

Derived lower bounds for the $v$-adic absolute value of linear forms.

Extended results to primes in residue classes.

Abstract

We study a linear form in the values of Euler's series $F(t)=\sum_{n=0}^\infty n!t^n$ at algebraic integer points $\alpha_1, \ldots, \alpha_m \in \mathbb{Z}_{\mathbb{K}}$ belonging to a number field $\mathbb{K}$. Let $v|p$ be a non-Archimedean valuation of $\mathbb{K}$. Two types of non-vanishing results for the linear form $\Lambda_v = \lambda_0 + \lambda_1 F_v(\alpha_1) + \ldots + \lambda_m F_v(\alpha_m)$, $\lambda_i \in \mathbb{Z}_{\mathbb{K}}$, are derived, the second of them containing a lower bound for the $v$-adic absolute value of $\Lambda_v$. The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Pad\'e approximations to the generalised factorial series $\sum_{n=0}^\infty \left( \prod_{k=0}^{n-1} P(k) \right) t^n$, where $P(x)$ is a polynomial of degree one.