Euler‘s factorial series at algebraic integer points

Abstract We study a linear form in the values of Euler’s series \(F(t)=\sum\nolimits_{n=0}^\infty n!t^n\) at algebraic integer points \(α_j∈\mathbb{Z}_\mathbb{K}, j=1,…,m\), belonging to a number field \(\mathbb{K}\). In the two main results it is shown that there exists a non-Archimedean valuation \(v\vert p\) of the field \(\mathbb{K}\) such that the linear form \({\mathrm\Lambda}_v=\lambda_0+\lambda_1F_v(\alpha_1)+\dots+\lambda_mF_v(\alpha_m)\), \(\lambda_i\in{\mathbb{Z}}_\mathbb{K}\), does not vanish. The second result contains a lower bound for the v-adic absolute value of \({\mathrm\Lambda}_v\), and the first one is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series \(\sum\nolimits_{n=0}^\infty{\left(\prod\nolimits_{k=0}^{n-1}P(k)\right)}t^n\), where \(P(x)\) is a polynomial of degree one