Further WZ-based methods for proving and generalizing Ramanujan's series. Issue 3 (4th March 2023)

• Record Type: Journal Article
• Authors: Campbell, John M.; Levrie, Paul
• Is Part Of: Journal of difference equations and applications. Volume 29:Issue 3(2023)
• Page Start: 366
• Page End: 376
• Publication Date: 2023-03-04
• Subjects: WZ method -- WZ theory -- Ramanujan-like series -- Ramanujan series -- hypergeometric series; 33F10; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2023.2198042
• Languages: English
• ISSNs: 1023-6198

Abstract

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 .