Applications via series accelerations of new identities involving Catalan-type numbers

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.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.