Applications of Zeilberger’s Algorithm to Ramanujan-Inspired Series Involving Harmonic-Type Numbers

Abstract

A “harmonic variant” of Zeilberger’s algorithm is utilized to improve upon the results introduced by Wang and Chu [ Ramanujan J. 52 (2020) 641–668]. Wang and Chu’s coefficient-extraction methodologies yielded evaluations for Ramanujan-like series involving summand factors of the form H 3 n +3 H n H (2) n +2 H (3) n , where H n denotes a harmonic number and H ( x ) n is a generalized harmonic number. However, it is unclear as to how Wang and Chu’s techniques could be applied to improve upon such results by separately evaluating the series obtained upon the expansion of the summands according to the terms of the factor H 3 n +3 H n H (2) n +2 H (3) n . In this note, we succeed in applying Zeilberger’s algorithm toward this problem, providing explicit evaluations for the series with a factor of the form H (3) n obtained from the aforementioned expansion. Our approach toward generalizing Zeilberger’s algorithm to non-hypergeometric expressions may be applied much more broadly. The series obtained by replacing H (3) n with H (2) n were highlighted as especially beautiful motivating examples in Wang and Chu’s article. These H (2) n -series motivate our main results, which are natural higher-order extensions of these H (2) n -series.