Euler's factorial series, Hardy integral, and continued fractions
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 Pad\'e approximations to prove a lower bound for the $p$-adic absolute value of the expression $cE_p\left(\pm p^a\right)-d$, where $c, d \in \mathbb{Z}$. It is interesting that the same Pad\'e 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 Pad\'e polynomial needed in the derivation of the lower bound for $|cE_p\left(\pm p^a\right)-d|_p$. Furthermore, we present an interconnection between $E(t)$ and $\mathcal{H}(t)$ via continued fractions.