Euler's factorial series, Hardy integral, and continued fractions

Abstract

We study $p$-adic Euler's series $E_p(t)={\sum }_{k=0}^{\infty }k!t^k$ at a point $p^a$, $a\in {\mathbb{Z}}_{\ge 1}$, and use Padé approximations to prove a lower bound for the $p$-adic absolute value of the expression $c{E}_p\left(\pm p^a\right)-d$, where $c,d\in \mathbb{Z}$. It is interesting that the same Padé polynomials which $p$-adically converge to $E_p(t)$, approach the Hardy integral $\mathcal{H}(t)={\int }_0^{\infty }\frac{e^{-s}}{1-ts}ds$ on the Archimedean side. This connection is used with a trick of analytic continuation when deducing an Archimedean bound for the numerator Padé polynomial needed in the derivation of the lower bound for $|c{E}_p\left(\pm p^a\right)-d{|}_p$. Furthermore, we present an interconnection between $E(t)$ and $\mathcal{H}(t)$ via continued fractions.