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 K\i l\i\c{c}, 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 ${}_{3}F_{2}(1)$-series with three real parameters from $1$ to $\frac{1}{4}$, and we apply this identity to generalize Ramanujan-like series for $\frac{1}{\pi}$, $\frac{\sqrt{2}}{\pi}$, $\frac{\sqrt{3}}{\pi}$, and $ \frac{\sqrt{2 \pm \sqrt{2}}}{\pi } $ that are due to Chu et al. A fast-converging series for $\pi^2$ 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 ${}_{3}F_{2}(1)$-family with three real parameters from $1$ to $\frac{1}{16}$, and we apply this result, via a series bisection, to formulate a new WZ proof of Ramanujan's series for $\frac{1}{\pi}$ of convergence rate $\frac{1}{4}$. A number of our finite 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.