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

We study p-adic Euler's series Ep(t)=∑k=0∞k!tk at a point pa, a∈Z≥1, and use Padé approximations to prove lower bound for the absolute value of expression cEp(±pa)−d, where c,d∈Z. It is interesting that same polynomials which p-adically converge Ep(t), approach Hardy integral H(t)=∫0∞e−s1−tsds on Archimedean side. This connection used with trick analytic continuation when deducing an numerator polynomial needed in derivation |cEp(±pa)−d|p. Furthermore, we present interconnection between E(t) H(t) via continued fractions.