Series acceleration formulas obtained from experimentally discovered hypergeometric recursions

TL;DR

This paper develops new hypergeometric recurrences using the WZ method, leading to series acceleration formulas for special constants like nd Ramanujan-type series for nd Catalan's constant, expanding on prior identities.

Contribution

It introduces a variety of hypergeometric recurrences and series acceleration identities, generalizing known series for nd Catalan's constant using the WZ method.

Findings

Derived new series acceleration formulas for nd Catalan's constant.

Generalized Ramanujan-type series for nd .

Proved hypergeometric recurrences using the WZ method.

Abstract

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}, and many related results.