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

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Date

2023-03

Department

Department of Mathematics and Systems Analysis

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Mcode

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Language

en

Pages

27

Series

Journal of Number Theory, Volume 244, pp. 224-250

Abstract

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 a lower bound for the p-adic absolute value of the expression cEp(±pa)−d, where c,d∈Z. It is interesting that the same Padé polynomials which p-adically converge to Ep(t), approach the Hardy integral [Formula presented] 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 |cEp(±pa)−d|p. Furthermore, we present an interconnection between E(t) and H(t) via continued fractions.

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Funding Information: The research of Ernvall-Hytönen and Seppälä was supported by the Emil Aaltonen Foundation . A big part of the work of Seppälä was conducted during her time at Aalto University supported by a grant from the Magnus Ehrnrooth Foundation . Publisher Copyright: © 2022 The Author(s)

Keywords

Continued fractions, Diophantine approximation, p-adic, Padé approximation

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DOI

10.1016/j.jnt.2022.09.007

Citation

Ernvall-Hytönen, A M, Matala-aho, T & Seppälä, L 2023, ' Euler's factorial series, Hardy integral, and continued fractions ', Journal of Number Theory, vol. 244, pp. 224-250 . https://doi.org/10.1016/j.jnt.2022.09.007

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https://urn.fi/URN:NBN:fi:aalto-202212146917

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[article-cris] Perustieteiden korkeakoulu / SCI