On Series Involving Cubed Catalan Numbers

TL;DR

This paper derives new series involving powers of Catalan numbers using binomial identities, generalizes a series for 1/π, and finds Ramanujan-like series for 1/π² and 1/π³.

Contribution

It introduces new series involving cubes and fourth powers of Catalan numbers and generalizes known series for 1/π and related constants.

Findings

Derived series involving cubes and fourth powers of Catalan numbers.

Established a generalization of Bauer's series for 1/π.

Obtained Ramanujan-like series for 1/π² and 1/π³.

Abstract

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for $1/\pi$ and obtain some Ramanujan-like series for $1/\pi^2$ and~$1/\pi^3$.