Euler's factorial series at algebraic integer points

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.