A single parameter Hermite-Pad\'e series representations for Ap\'ery's constant

Anier Soria-Lorente; Stefan Berres

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arXiv · 1806.05988

A single parameter Hermite-Pad\'e series representations for Ap\'ery's constant

Anier Soria-Lorente·Stefan Berres

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Abstract

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

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Anier Soria-Lorente, Stefan Berres. 2018-06-15. A single parameter Hermite-Pad\'e series representations for Ap\'ery's constant. https://arxiv.org/abs/1806.05988

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