A single parameter Hermite–Padé series representation for Apéry’s constant

Anier Soria-Lorente and Stefan Berres
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 107—134
DOI: 10.7546/nntdm.2020.26.3.107-134
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Authors and affiliations

Anier Soria-Lorente
Department of Technology, University of Granma
Bayamo-Granam, Cuba

Stefan Berres
Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingeniería
Universidad Católica de Temuco, Temuco, Chile

Abstract

Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for ζ(3) which only depends on one single integer parameter. This is accomplished by deducing a Hermite–Padé approximation problem using ideas of Sorokin (1998). As a consequence we get a new recurrence relation for the approximation of ζ(3) as well as a corresponding new continued fraction expansion for ζ(3), which do no reproduce Apéry’s phenomenon, i.e., though the approaches are different, they lead to the same sequence of Diophantine approximations to ζ(3). Finally, the convergence rates of several series representations of ζ(3) are compared.

Keywords

  • Riemann zeta function
  • Apéry’s theorem
  • Hermite–Padé approximation problem
  • Recurrence relation
  • Continued fraction expansion
  • Series representation

2010 Mathematics Subject Classification

  • Primary
    • 11B37
    • 30B70
    • 14G10
    • 11J72
    • 11M06
  • Secondary
    • 37B20
    • 11A55
    • 11J70
    • 11Y55
    • 11Y65

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Soria-Lorente, A., & Berres, S. (2020). A single parameter Hermite–Padé series representation for Apéry’s constant. Notes on Number Theory and Discrete Mathematics, 26 (3), 107-134, doi: 10.7546/nntdm.2020.26.3.107-134.

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