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Accelerating Dougall's $ _5F_4$-sum and infinite series involving $ \pi$


Authors: Wenchang Chu and Wenlong Zhang
Journal: Math. Comp. 83 (2014), 475-512
MSC (2010): Primary 33D15; Secondary 05A15
Published electronically: April 26, 2013
MathSciNet review: 3120601
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Abstract: The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall's $ _5H_5$-series. Several unusual transformation formulae into fast convergent series are established. They lead surprisingly to numerous infinite series identities involving $ \pi $, $ \zeta (3)$ and the Catalan constant, including several important ones discovered by Ramanujan (1914) and recently by Guillera.


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Additional Information

Wenchang Chu
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Address at time of publication: Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via Arnesano P. O. Box 193, 73100 Lecce, Italia
Email: chu.wenchang@unisalento.it

Wenlong Zhang
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: wenlong.dlut@yahoo.com.cn

DOI: https://doi.org/10.1090/S0025-5718-2013-02701-9
Keywords: Abel's lemma on summation by parts, classical hypergeometric series, partial sum of Dougall's $_5H_5$-series, acceleration of convergent series, $\pi$-formulae of Ramanujan and Guillera
Received by editor(s): December 9, 2011
Received by editor(s) in revised form: March 27, 2012
Published electronically: April 26, 2013
Article copyright: © Copyright 2013 American Mathematical Society