A single parameter Hermite-Pade series representation for Apery's constant

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Soria-Lorente, Anier
Berres, Stefan
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NOTES ON NUMBER THEORY AND DISCRETE MATHEMATICS,Vol.26,107-134,2020
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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 (3) which only depends on one single integer parameter. This is accomplished by deducing a Hermite-Pade 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 Apery's phenomenon, i.e., though the approaches are different, they lead to the same sequence of Diophantine approximations to zeta (3). Finally, the convergence rates of several series representations of zeta (3) are compared.
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