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Dougall's bilateral $ _2H_2$-series and Ramanujan-like $ \pi$-formulae


Author: Wenchang Chu
Journal: Math. Comp. 80 (2011), 2223-2251
MSC (2010): Primary 33C20; Secondary 40A25, 65B10
DOI: https://doi.org/10.1090/S0025-5718-2011-02474-9
Published electronically: March 2, 2011
MathSciNet review: 2813357
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Abstract: The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall's bilateral $ _2H_2$-series. Several unusual transformations into fast convergent series are established. They lead surprisingly to numerous infinite series expressions for $ \pi$, including several formulae discovered by Ramanujan (1914) and recently by Guillera (2008).


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Additional Information

Wenchang Chu
Affiliation: Hangzhou Normal University, Institute of Combinatorial Mathematics, Hangzhou 310036, People’s Republic of China
Address at time of publication: Dipartimento di Matematica, Università del Salento, Lecce–Arnesano, P. O. Box 193, Lecce 73100 Italy
Email: chu.wenchang@unisalento.it

DOI: https://doi.org/10.1090/S0025-5718-2011-02474-9
Keywords: Abel’s lemma on summation by parts, Classical hypergeometric series, The Gauss summation theorem, Dougall’s bilateral $_{2}H_{2}$-series identity, Ramanujan’s $\pi$-formulae
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: July 31, 2010
Published electronically: March 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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