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A single parameter Hermite-Pad\'e series representations for Ap\'ery's constant
Inspired by the results of Rhin and Viola (2001), the purpose of this work is
to elaborate on a series representation for 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
as well as a corresponding new continued fraction expansion for
, 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 . Finally, the convergence rates of
several series representations of are compared