Three-parameter generalizations of formulas due to Guillera

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.