Infinite series of convergence rate −1/4 about generalized harmonic numbers

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Abstract

Chu and Zhang (2014) found several remarkable transformations for well-poised hypergeometric series. One of them is employed in this paper to investigate infinite series of convergence rate “−1/4” involving generalized harmonic numbers. Numerous closed-form formulae are established that explicitly express a large class of series in terms of π, ln 2, and the Hurwitz zeta function. Thirteen transformations are derived from harmonic series into nine seemingly simpler multifold k-sums, whose exact values are still not determined.

Keywords

harmonic number / Gamma function / hypergeometric series / Hurwitz zeta function

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References

[1] E. D. Rainville, Special Functions , The Macmillan Company, 1960.

[2] W. Chu, Hypergeometric series and the Riemann zeta function, Acta Arithmetica, 82 (1997), 103-118. https://doi.org/10.4064/aa-82-2-103-118

[3] W. Chu, Hypergeometric approach to Apéry-like series, Integr. Transforms Spec. Funct., 28 (2017), 505-518.

[4] C. L. Li, W. Chu, Series of convergence rate −1/4 containing harmonic numbers, Axioms, 12 (2023), 513. https://doi.org/10.3390/axioms12060513

[5] C. L. Li, W. Chu, Gauss’ second theorem for 2 F₁( 1/2)-series and novel harmonic series identities , Mathematics, 12 (2024), 1381. https://doi.org/10.3390/math12091381

[6] L. Comtet, Advanced Combinatorics , Springer Dordrecht, 1974. https://doi.org/10.1007/978-94-010-2196-8

[7] K. Adegoke, R. Frontczak, T. Goy, Combinatorial sums, series and integrals involving odd harmonic numbers, Afr. Mat., 36 (2025), 124. https://doi.org/10.1007/s13370-025-01346-1

[8] W. Chu, Ramanujan-like formulae for π^±1 via Gould-Hsu inverse series relations , Ramanujan J., 56 (2021), 1007-1027. https://doi.org/10.1007/s11139-020-00337-z

[9] X. Y. Wang, W. Chu, Series with harmonic-like numbers and squared binomial coefficients, Rocky Mt. J. Math., 52 (2022), 1849-1866. https://doi.org/10.1216/rmj.2022.52.1849

[10] K. W. Chen, Hypergeometric series and generalized harmonic numbers, J. Differ. Equations Appl., 31 (2025), 85-114. https://doi.org/10.1080/10236198.2024.2388746

[11] W. Chu, W. L. Zhang, Accelerating Dougall’s 5 F₄-sum and infinite series involving π , Math. Comput., 83 (2014), 475-512. https://doi.org/10.1090/S0025-5718-2013-02701-9

[12] W. Chu, J. M. Campbell, Harmonic sums from the Kummer theorem, J. Math. Anal. Appl., 501 (2021), 125179. https://doi.org/10.1016/j.jmaa.2021.125179

[13] W. N. Bailey, Generalized Hypergeometric Series , Cambridge University Press, 1935.

[14] K. N. Boyadzhiev, Series with central binomial coefficients, Catalan numbers, and harmonic numbers, J. Integer Sequences, 15 (2012), 7.

[15] W. Chu, Alternating series of Apéry-type for the Riemann zeta function, Contrib. Discrete Math., 15 (2020), 108-116. https://doi.org/10.55016/ojs/cdm.v15i3.62721

[16] W. Chu, Further Apéry-like series for Riemann zeta function, Math. Notes, 109 (2021), 136-146. https://doi.org/1134/S0001434621010168

[17] Q. H. Hou, Z. W. Sun, Taylor coefficients and series involving harmonic numbers, preprint, arXiv: 2310.03699. https://doi.org/10.48550/arXiv.2310.03699

[18] X. Wang, W. Chu, Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients, Ramanujan J., 52 (2020), 641-668. https://doi.org/10.3390/math11139-019-00140-5

[19] Z. W. Sun, New series involving binomial coefficients, preprint, arXiv:2307.03086. https://doi.org/10.48550/arXiv.2307.03086

[20] K. H. Pilehrood, T. H. Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, Electron. J. Combin., 18 (2012), P35. https://doi.org/10.37236/2049

[21] Z. W. Sun, Series with summands involving higher harmonic numbers, in Combinatorial and Additive Number Theory VI, (eds. M. B. Nathanson), Springer, (2025), 413-460. https://doi.org/10.1007/978-3-031-65064-221

[22] A. S. Olaikhan, An Introduction To the Harmonic Series And Logarithmic Integrals: For High School Students Up To Researchers , Self-published, Phoenix, AZ, 2023.