A rapidly converging Ramanujan-type series for Catalan's constant

TL;DR

This paper introduces new Ramanujan-type series for Catalan's constant with faster convergence rates than existing formulas, potentially enabling novel irrationality proofs.

Contribution

It presents two new rapidly converging hypergeometric series for Catalan's constant derived from known identities, surpassing previous series in convergence speed.

Findings

New series converge faster than classical formulas

Potential for Apéry-like irrationality proof for Catalan's constant

Improved computational efficiency for evaluating Catalan's constant

Abstract

In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Ap\'{e}ry-like irrationality proof could be found for this constant.