A WZ proof for a Ramanujan-like series involving cubed binomial coefficients

Abstract

We make use of a cubic analogue of telescoping sums for WZ pairs, in order to prove an evaluation involving for a Ramanujan-like series containing odd harmonic numbers and third powers of Catalan numbers as summand factors. Closed-form evaluations for hypergeometric series expressions for complete elliptic integrals play a key role in our proof. While our ‘cubic WZ’ method is of interest in its own right, we also consider how summation methods given by Ablinger may be applied to obtain an alternate algorithmic proof of our main result.