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ON GUILLERA’S
${}_{7}F_{6}( \frac {27}{64} )$-SERIES FOR 
${1}/{\pi ^2}$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 09 February 2023, pp. 464-471
- Print publication:
- December 2023
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- Article
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In 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 
$\sqrt {2}$.
