Further WZ-based methods for proving and generalizing Ramanujan's series. Issue 3 (4th March 2023) - Campbell, John M.; Levrie, Paul
ABSTRACT: In 2002 and 2006, using a Wilf–Zeilberger-based method, Guillera introduced proofs for evaluations for what are considered as the simplest two series out of Ramanujan's 17 series for 1 π . In this article, we show how the WZ method may be used in a fundamentally and nontrivially different way to prove these results, and to obtain identities for infinite families of Ramanujan-like series for 1 π . We introduce a 3 F 2 -recurrence that we had discovered experimentally, and we prove this recursion using the WZ method and apply it to obtain a series acceleration formula that we apply to formulate a new and simple proof for the Ramanujan series for 1 π that has a convergence rate of 1 64, and we provide an infinite family of generalizations of this formula, and similarly for Ramanujan's series of convergence rate 1 4 .