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

TL;DR

This paper introduces a novel single-parameter Hermite-Padé series representation for Apéry's constant, providing new recurrence relations and continued fraction expansions that match Apéry's approximations.

Contribution

It develops a new single-parameter Hermite-Padé approximation framework for b6(3), leading to fresh recurrence relations and continued fractions distinct from previous methods.

Findings

Derives a new recurrence relation for b6(3)

Establishes a new continued fraction expansion for b6(3)

Compares convergence rates of various series representations

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.