Mathematical Research Letters

Volume 16 (2009)

Number 3

The Apéry numbers, the Almkvist-Zudilin numbers and new series for $1/\pi$

Pages: 405 – 420



Heng Huat Chan (National University of Singapore)

Helena Verrill (Louisiana State University)


This paper concerns series for $1/\pi$, such as Sato’s series (\ref{eqn:sato}), and the series of H.H. Chan, S.H. Chan and Z.-G. Liu (\ref{eqn:exampleforb_ks}) below.The examples of Sato, Chan, Chan and Liu are related to two index $2$ subgroups of $\Gamma_0(6)_{+}$. These examples motivate us to look at a third subgroup of $\Gamma_0(6)_{+}$. {We give a new method of constructing such series using the theory of modular forms and conclude our work with several new examples.}

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