Euler's factorial series, Hardy integral, and continued fractions (Q2112782)

Let function \(E_p(t)\) be defined by Euler series \[ E_p(t):=\sum_{k=0}^\infty k! t^k \] which converges in the \(p\)-adic metric \(|p|_p=p^{-1}\) for a prime \(p\) in the interior of disk \(\{ t \in \mathbb{C}_p\, \mid\, |t|_p <p^{\frac{1}{p-1}}\}\). Let \[ \mathcal{H}(t)=\int_0^\infty \frac{e^{-s}}{1-ts} ds \] be the Hardy integral. The \(p\)-adic lower bounds for the linear form \(cE_p(\pm p^a)-d\) with integer coefficients \(c,d\) and a positive integer \(a\) are getting in the paper. The following result is proven using the Padé approximation. Let \(p\) be a prime number and \(H \in \mathbb{Z}_{\geq 4}\). Suppose that \(a\) is a positive integer such that \(p^a >c_1 \log (c_2 H),\) where \(c_1, c_2\) are constants. Then, for all \(c,d \in \mathbb{Z}, c\ne 0\) with \(|c|+|d| \leq H\) there holds \[ |cE_p(\pm p^a)-d|_p > \Big(2He^{\frac{11}{16}}\Big)^{-\frac{32}{11} a \log p}. \] An interconnection between \(E(t)\) and \(\mathcal{H}(t)\) via continued fractions is presented.