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 -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.
