Three-parameter generalizations of formulas due to Guillera
John M. Campbell
TL;DR
The paper addresses extending Guillera's $1/\pi^2$ series by a three-parameter acceleration not equivalent to Chu–Zhang's, enabling new hypergeometric series with Guillera-like convergence rates. It develops a double-acceleration framework based on Zeilberger's algorithm applied to a bivariate summand $F(n,k)$, yielding a telescoping identity and a recursion for $f(n)=\sum_k F(n,k)$, with special parameter choices recovering Guillera's formulas and known Chu–Zhang results while producing new series. The principal contribution is Theorem maintheorem, which provides a three-parameter generalization of Guillera's $1/\pi^2$ formula, along with explicit new series (including rate $-\frac{1}{1024}$ instances) and connections to other constants such as $\zeta(3)$. This work broadens the toolkit for rapidly convergent Ramanujan-type series and offers alternative avenues for symbolic proof and high-precision computation in hypergeometric summation.
Abstract
Guillera has introduced remarkable series expansions for $\frac{1}{π^2}$ 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 $\frac{1}{π^2}$ that are of the same convergence rates as Guillera's series and that have not previously been known.