A single parameter Hermite-Padé series representations for Apéry's constant

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-Padé 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 Apéry'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.