Applications Via Series Accelerations of New Identities Involving Catalan-Type Numbers

Abstract

We introduce infinite families of terminating hypergeometric identities involving generalizations of Catalan numbers, generalizing results introduced by Chu and Kılıç, and we apply Wilf–Zeilberger (WZ) pairs associated with our new identities via a series acceleration method. We apply a WZ pair introduced in our article to prove an identity for accelerating the convergence for a family of <inf>3</inf>F<inf>2</inf>(1)-series with three real parameters from 1 to 1/4, and we apply this identity to generalize Ramanujan-like series for (Formula presented) that are due to Chu et π π π al. A fast-converging series for π2 due to Guillera is also a special case of our acceleration identity. We also apply another WZ pair introduced in this article to prove an identity for accelerating the convergence of a<inf>3</inf>F<inf>2</inf>(1)-family with three real parameters from 1 to 1/16 , and we apply 16 this result, via a series bisection, to formulate a new WZ proof of Ramanujan’s series for of convergence rate 1/4. A number of our finite π 4 sums involving Catalan-type numbers are such that up-to-date versions of the Maple Computer Algebra System cannot compute WZ pairs for such sums, which is representative of the computationally challenging nature of our results. © 2025, University of Calgary. All rights reserved.