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Article contents
ON GUILLERA’S 
${}_{7}F_{6}( \frac {27}{64} )$-SERIES FOR 
${1}/{\pi ^2}$
Published online by Cambridge University Press: 09 February 2023
Abstract
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 
$\sqrt {2}$.
MSC classification
Secondary:
65B10: Summation of series
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 108 , Issue 3 , December 2023 , pp. 464 - 471
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935).Google Scholar
Beukers, F. and Forsgøard, J., ‘
$\varGamma$
-evaluations of hypergeometric series’, Ramanujan J. 58 (2022), 677–699.10.1007/s11139-022-00566-4CrossRefGoogle Scholar
$\varGamma$
-evaluations of hypergeometric series’, Ramanujan J. 58 (2022), 677–699.10.1007/s11139-022-00566-4CrossRefGoogle ScholarCampbell, J. M., ‘WZ proofs of identities from Chu and Kılıç, with applications’, Appl. Math. E-Notes 22 (2022), 354–361.Google Scholar
Chen, X. and Chu, W., ‘
$q$
-analogues of Guillera’s two series for
${\pi}^{\pm 2}$
with convergence rate
$\frac{27}{64}$
’, Int. J. Number Theory 17 (2021), 71–90.10.1142/S1793042121500056CrossRefGoogle Scholar
$q$
-analogues of Guillera’s two series for
${\pi}^{\pm 2}$
with convergence rate
$\frac{27}{64}$
’, Int. J. Number Theory 17 (2021), 71–90.10.1142/S1793042121500056CrossRefGoogle ScholarChu, W., ‘Terminating
${}_2{F}_1(4)$
-series perturbed by two integer parameters’, Proc. Amer. Math. Soc. 145 (2017), 1031–1040.10.1090/proc/13293CrossRefGoogle Scholar
${}_2{F}_1(4)$
-series perturbed by two integer parameters’, Proc. Amer. Math. Soc. 145 (2017), 1031–1040.10.1090/proc/13293CrossRefGoogle ScholarChu, W., ‘Infinite series formulae related to Gauss and Bailey -sums’, Acta Math. Sci. Ser. B (Engl. Ed.) 40 (2020), 293–315.Google Scholar
Chu, W., ‘Seven hundreds of exotic
${}_3{F}_2$
-series evaluated in
$\pi$
,
$\sqrt{2}$
and
$\log (1+\sqrt{2})$
’, Contemp. Math. 2 (2021), 327–398.10.37256/cm.2420211066CrossRefGoogle Scholar
${}_3{F}_2$
-series evaluated in
$\pi$
,
$\sqrt{2}$
and
$\log (1+\sqrt{2})$
’, Contemp. Math. 2 (2021), 327–398.10.37256/cm.2420211066CrossRefGoogle ScholarChu, W. and Kılıç, E., ‘Binomial sums involving Catalan numbers’, Rocky Mountain J. Math. 51 (2021), 1221–1225.10.1216/rmj.2021.51.1221CrossRefGoogle Scholar
Chu, W. and Zhang, W., ‘Accelerating Dougall’s
${}_5{F}_4$
-sum and infinite series involving
$\pi$
’, Math. Comput. 83 (2014), 475–512.10.1090/S0025-5718-2013-02701-9CrossRefGoogle Scholar
${}_5{F}_4$
-sum and infinite series involving
$\pi$
’, Math. Comput. 83 (2014), 475–512.10.1090/S0025-5718-2013-02701-9CrossRefGoogle ScholarEbisu, A., ‘On a strange evaluation of the hypergeometric series by Gosper’, Ramanujan J. 32 (2013), 101–108.10.1007/s11139-013-9483-1CrossRefGoogle Scholar
Ebisu, A., ‘Special values of the hypergeometric series’, Mem. Amer. Math. Soc. 248 (2017), 1–96.Google Scholar
Ekhad, S. B., ‘Forty “strange” computer-discovered [and computer-proved (of course!)] hypergeometric series evaluations’, Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, 2004. https://sites.math.rutgers.edu/zeilberg/mamarim/mamarimhtml/strange.html.Google Scholar
Guillera, J., ‘Some binomial series obtained by the WZ-method’, Adv. Appl. Math. 29 (2002), 599–603.10.1016/S0196-8858(02)00034-9CrossRefGoogle Scholar
Guillera, J., ‘Generators of some Ramanujan formulas’, Ramanujan J. 11 (2006), 41–48.10.1007/s11139-006-5306-yCrossRefGoogle Scholar
Guillera, J., ‘A new method to obtain series for
$1/\pi$
and
$1/ {\pi}^2$
’, Exp. Math. 15 (2006), 83–89.10.1080/10586458.2006.10128943CrossRefGoogle Scholar
$1/\pi$
and
$1/ {\pi}^2$
’, Exp. Math. 15 (2006), 83–89.10.1080/10586458.2006.10128943CrossRefGoogle ScholarGuillera, J., ‘Hypergeometric identities for 10 extended Ramanujan-type series’, Ramanujan J. 15 (2008), 219–234.10.1007/s11139-007-9074-0CrossRefGoogle Scholar
Guillera, J., ‘On WZ-pairs which prove Ramanujan series’, Ramanujan J. 22 (2010), 249–259.10.1007/s11139-010-9238-1CrossRefGoogle Scholar
Guillera, J., ‘A new Ramanujan-like series for
$1/ {\pi}^2$
’, Ramanujan J. 26 (2011), 369–374.10.1007/s11139-010-9259-9CrossRefGoogle Scholar
$1/ {\pi}^2$
’, Ramanujan J. 26 (2011), 369–374.10.1007/s11139-010-9259-9CrossRefGoogle ScholarGuillera, J., ‘Dougall’s
${}_5{F}_4$
sum and the WZ algorithm’, Ramanujan J. 46 (2018), 667–675.10.1007/s11139-018-9998-6CrossRefGoogle Scholar
${}_5{F}_4$
sum and the WZ algorithm’, Ramanujan J. 46 (2018), 667–675.10.1007/s11139-018-9998-6CrossRefGoogle ScholarMao, G.-S. and Tauraso, R., ‘Three pairs of congruences concerning sums of central binomial coefficients’, Int. J. Number Theory 17 (2021), 2301–2314.10.1142/S1793042121500895CrossRefGoogle Scholar
Marichev, O., Sondow, J. and Weisstein, E. W., ‘Catalan’s Constant’, from MathWorld–A Wolfram Web Resource, 2022. https://mathworld.wolfram.com/CatalansConstant.html.Google Scholar
Oerlemans, A. H. M., The Cupproduct’s Applications to Ebisu’s Method, Master Thesis, Universiteit Utrecht, 2019.Google Scholar
Petkovšek, M., Wilf, H. S. and Zeilberger, D.,
$A=B$
(A. K. Peters, Ltd., Wellesley, MA, 1996).Google Scholar
$A=B$
(A. K. Peters, Ltd., Wellesley, MA, 1996).Google ScholarWeisstein, E. W., ‘Apéry’s Constant’, from MathWorld–A Wolfram Web Resource, 2022. https://mathworld.wolfram.com/AperysConstant.html.Google Scholar
Weisstein, E. W., ‘Pi Squared’, from MathWorld–A Wolfram Web Resource, 2022. https://mathworld.wolfram.com/PiSquared.html.Google Scholar
Weisstein, E. W., ‘Pythagoras’s constant’, from MathWorld–A Wolfram Web Resource, 2022. https://mathworld.wolfram.com/PythagorassConstant.html.Google Scholar