Accelerating Dougall's \(_{5}F_{4}\)-sum and infinite series involving \(\pi\) (Q2862541)

In previous publications by the first author, a modified version of Abel's lemma on summation by parts was used to obtain several summation and transformation formulas involving classical and basic hypergeometric series. In this paper the modified lemma is applied to the bilateral well-poised \({}_5H_5\)-series due to Dougall. This leads to three fundamental recurrence relations. Iterations of these lead to several transformation formulas for Dougall's \({}_5H_5\)-series. These transformation formulas further lead to numerous formulas for \(\pi\) in terms of fast convergent infinite series, including severel ones earlier discovered by Ramanujan (1914) and recently by Guillera. Furthermore, fast convergent series are obtained for \(\zeta(3)\) and Catalan's constant \(G\). The authors make a careful selection of 125 explicit examples of fast convergent infinite series for \(\pi\), \(\zeta(3)\) and \(G\).