Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$dE_p\left(p^a\right)-c$, 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 $\mathcal{H}(t) = \int_0^\infty
\frac{e^{-s}}{1-ts}\mathrm{d}s$ 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 $|dE_p\left(p^a\right)-c|_p$. Furthermore, we present an interconnection
between $E(t)$ and $\mathcal{H}(t)$ via continued fractions.

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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
1
The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: stat