Accelerating Dougall’s ₅𝐹₄-sum and infinite series involving 𝜋

Chu, Wenchang; Zhang, Wenlong

doi:10.1090/S0025-5718-2013-02701-9

Accelerating Dougall’s $_5F_4$-sum and infinite series involving $\pi$
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Math. Comp. 83 (2014), 475-512 Request permission

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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