ON GUILLERA’S -SERIES FOR

Abstract

AbstractIn 2011, Guillera [‘A new Ramanujan-like series for $1/\pi ^2$ ’, Ramanujan J.26 (2011), 369–374] introduced a remarkable rational ${}_{7}F_{6}( \frac {27}{64} )$ -series for ${1}/{\pi ^2}$ using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall’s ${}_{5}F_{4}$ -sum. Another proof of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$ -series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall’s sum together with the WZ method. Subsequently, Chen and Chu introduced a q-analogue of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$ -series. The many past research articles concerning Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$ -series for ${1}/{\pi ^2}$ naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational ${}_{7}F_{6}( \frac {27}{64} )$ - and ${}_{6}F_{5}( \frac {27}{64} )$ -series for $\sqrt {2}$ .