A summation formula and Ramanujan type series

From Wallis and Forsyth to Ramanujan

“…Series for 1 π have received much attention lately, see for instance the survey paper by Baruah, Berndt and Chan [2] and the work of Guillera [7,8] who uses the Wilf-Zeilberger method to find series of this form. Chu [3] and Liu [11,12] have obtained similar results by manipulating hypergeometric series using Dougall's and Gauss's summation formulae.…”

“…By combining the two recurrences ( 9) and ( 11) we can derive other series. For instance, if we use (11) again in the recurrence of Theorem 3, we get this series (+2, +1-scheme):…”

π-Formulas with free parameters

“…Recently, Chu [5,6] and Liu [7,8] have deduced many surprising π -formulas from some known hypergeometric series identities. Thereinto, Chu [5] showed that (5) implies the Ramanujan-type series for 1/π with three free parameters:…”

“…Eqs. (6)- (8) can create numerous special π -formulas by specifying the free parameters. For example, the case m = n = p = 0 of (6) produces the simplest Ramanujan-type formula due to Bauer [9, Section 4] (see also [10]):…”

“…Seven general π -formulas with free parameters, which include (1)-(3) as special cases, will be derived from (4) in Section 2. Three general π -formulas with free parameters, which include (6)- (8) as special cases, will be deduced from (5) in Section 3.…”