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Published online by Cambridge University Press:  07 July 2023

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou (Henan), PR China e-mail:
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou (Henan), PR China


By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun [‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7]. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.

Research Article
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935).Google Scholar
Borwein, J. M. and Borwein, P. B., Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1987).Google Scholar
Chan, H. H. and Liaw, W. C., ‘Cubic modular equations and new Ramanujan-type series for $1/ \pi$ ’, Pacific J. Math. 192 (2000), 219238.CrossRefGoogle Scholar
Chen, X. and Chu, W., ‘Dixon’s ${}_3{F}_2(1)$ -series and identities involving harmonic numbers and Riemann zeta function’, Discrete Math. 310(1) (2010), 8391.CrossRefGoogle Scholar
Chu, W., ‘Hypergeometric series and the Riemann Zeta function’, Acta Arith. 82(2) (1997), 103118.CrossRefGoogle Scholar
Chu, W., ‘Dougall’s bilateral ${}_2{H}_2$ -series and Ramanujan-like $\pi$ -formulae’, Math. Comp. 80(276) (2011), 22232251.CrossRefGoogle Scholar
Chu, W., ‘Hypergeometric approach to Apéry-like series’, Integral Transforms Spec. Funct. 28(7) (2017), 505518.CrossRefGoogle Scholar
Chu, W., ‘Infinite series identities from the very-well-poised $\varOmega$ -sum’, Ramanujan J. 55(1) (2021), 239270.CrossRefGoogle Scholar
Chu, W., ‘Further Apéry-like series for Riemann zeta function’, Math. Notes 109(1) (2021), 136146.CrossRefGoogle Scholar
Chu, W. and Campbell, J. M., ‘Harmonic sums from the Kummer theorem’, J. Math. Anal. Appl. 501(2) (2021), Article no. 125179, 37 pages.CrossRefGoogle Scholar
Chu, W. and Zhang, W. L., ‘Accelerating Dougall’s ${}_5{F}_4$ -sum and infinite series involving $\pi$ ’, Math. Comp. 83(285) (2014), 475512.CrossRefGoogle Scholar
Comtet, L., Advanced Combinatorics (Dordrecht–Holland, The Netherlands, 1974).CrossRefGoogle Scholar
Elsner, C., ‘On sums with binomial coefficient’, Fibonacci Quart. 43(1) (2005), 3145.Google Scholar
Guillera, J., ‘About a new kind of Ramanujan-type series’, Exp. Math. 12(4) (2003), 507510.CrossRefGoogle Scholar
Guillera, J., ‘Generators of some Ramanujan formulas’, Ramanujan J. 11(1) (2006), 4148.CrossRefGoogle Scholar
Guillera, J., ‘Hypergeometric identities for 10 extended Ramanujan-type series’, Ramanujan J. 15(2) (2008), 219234.CrossRefGoogle Scholar
Lehmer, D. H., ‘Interesting series involving the central binomial coefficient’, Amer. Math. Monthly 92 (1985), 449457.CrossRefGoogle Scholar
Rainville, E. D., Special Functions (The Macmillan Company, New York, 1960).Google Scholar
Ramanujan, S., ‘Modular equations and approximations to $\pi$ ’, Quart. J. Math. (Oxford) 45 (1914), 350372.Google Scholar
Sun, Z.-W., ‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7.Google Scholar
Wang, X. Y. and Chu, W., ‘Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients’, Ramanujan J. 52(3) (2020), 641668.CrossRefGoogle Scholar
Zucker, I. J., ‘On the series ${\sum}_{k=1}^{\infty }{\left(\genfrac{}{}{0pt}{}{2k}{k}\right)}^{-1}{k}^{-n}$ ’, J. Number Theory 20(1) (1985), 92102.CrossRefGoogle Scholar