Three-parameter generalizations of formulas due to Guillera

TL;DR

This paper generalizes Guillera's series formulas for 1/π^2 using an acceleration method based on Zeilberger's algorithm, resulting in new hypergeometric series with similar convergence rates.

Contribution

It introduces three-parameter generalizations of Guillera's formulas for 1/π^2 using an acceleration technique related to Zeilberger's algorithm.

Findings

Derived new hypergeometric series for 1/π^2

Achieved series with convergence rates of -1/1024 and -1/4

Expanded the set of known series for 1/π^2

Abstract

Guillera has introduced remarkable series expansions for $\frac{1}{\pi^2}$ of convergence rates $-\frac{1}{1024}$ and $-\frac{1}{4}$ via the Wilf-Zeilberger method. Through an acceleration method based on Zeilberger's algorithm and related to Chu and Zhang's series accelerations based on Dougall's ${}_{5}H_{5}$-series, we introduce and prove three-parameter generalizations of Guillera's formulas. We apply our method to construct rational, hypergeometric series for $\frac{1}{\pi^2}$ that are of the same convergence rates as Guillera's series and that have not previously been known.