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

2021/11/26 by Anne-Maria Ernvall-Hytönen, Ernvall-Hytönen, Anne-Maria, Tapani Matala-aho +3
Mathematics · #advanced mathematical theories #Meromorphic and Entire Functions #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2111.13649

Abstract

We study p-adic Euler's series E_p(t) = ∑_k=0^∞k!t^k at a point p^a, a ∈ ℤ_{≥ 1}, and use Padé approximations to prove a lower bound for the p-adic absolute value of the expression cE_p(± p^a)-d, where c, d ∈ ℤ. It is interesting that the same Padé polynomials which p-adically converge to E_p(t), approach the Hardy integral H(t) = ∫₀^∞ \frace^-s1-tsds 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 |cE_p(± p^a)-d|_p. Furthermore, we present an interconnection between E(t) and H(t) via continued fractions.

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