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.
Categories: math.CA