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

Anne Maria Ernvall-Hytönen; Tapani Matala-aho; Louna Seppälä

doi:10.1016/j.jnt.2022.09.007

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

Anne Maria Ernvall-Hytönen, Tapani Matala-aho^*, Louna Seppälä

^*Tämän työn vastaava kirjoittaja

University of Helsinki

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

5 Sitaatiot (Scopus)

85 Lataukset (Pure)

Abstrakti

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 [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 |cE_p(±p^a)−d|_p. Furthermore, we present an interconnection between E(t) and H(t) via continued fractions.

Alkuperäiskieli	Englanti
Sivut	224-250
Sivumäärä	27
Julkaisu	Journal of Number Theory
Vuosikerta	244
DOI - pysyväislinkit	https://doi.org/10.1016/j.jnt.2022.09.007
Tila	Julkaistu - maalisk. 2023
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Pääsy asiakirjaan

10.1016/j.jnt.2022.09.007Lisenssi: CC BY

SCI_Ernvall-Hytönen_etal_Journal_of_Number_Theory_2022Final published version, 419 KBLisenssi: CC BY

Muut tiedostot ja linkit

Linkki julkaisuun Scopuksessa

Siteeraa tätä

@article{a0ace7f15f3c4ebf95133a9e2a7610ac,

title = "Euler's factorial series, Hardy integral, and continued fractions",

abstract = "We study p-adic Euler's series Ep(t)=∑k=0∞k!tk at a point pa, a∈Z≥1, and use Pad{\'e} 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 Pad{\'e} 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 Pad{\'e} 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.",

keywords = "Continued fractions, Diophantine approximation, p-adic, Pad{\'e} approximation",

author = "Ernvall-Hyt{\"o}nen, {Anne Maria} and Tapani Matala-aho and Louna Sepp{\"a}l{\"a}",

note = "Funding Information: The research of Ernvall-Hyt{\"o}nen and Sepp{\"a}l{\"a} was supported by the Emil Aaltonen Foundation . A big part of the work of Sepp{\"a}l{\"a} was conducted during her time at Aalto University supported by a grant from the Magnus Ehrnrooth Foundation . Publisher Copyright: {\textcopyright} 2022 The Author(s)",

year = "2023",

month = mar,

doi = "10.1016/j.jnt.2022.09.007",

language = "English",

volume = "244",

pages = "224--250",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Elsevier",

}

TY - JOUR

T1 - Euler's factorial series, Hardy integral, and continued fractions

AU - Ernvall-Hytönen, Anne Maria

AU - Matala-aho, Tapani

AU - Seppälä, Louna

N1 - 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)

PY - 2023/3

Y1 - 2023/3

N2 - 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.

AB - 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.

KW - Continued fractions

KW - Diophantine approximation

KW - p-adic

KW - Padé approximation

UR - http://www.scopus.com/inward/record.url?scp=85140752224&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2022.09.007

DO - 10.1016/j.jnt.2022.09.007

M3 - Article

AN - SCOPUS:85140752224

SN - 0022-314X

VL - 244

SP - 224

EP - 250

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

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

Abstrakti

Pääsy asiakirjaan

Muut tiedostot ja linkit

Sormenjälki

Siteeraa tätä