Journal of Applied Mathematics and Computation

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Article http://dx.doi.org/10.26855/jamc.2023.03.008

Harmonic Number Identities from Log-integral Transformation

Kunzhen Zhang*, Xinhua Xiong

Department of Mathematics and Three Gorges Mathematics Research Center, China Three Gorges University, Yichang, Hubei, China.

*Corresponding author: Kunzhen Zhang

Published: April 13,2023

Abstract

The research on the summation of harmonic numbers and generalized harmonic numbers has a long history. In general, the summation formulae of infinite series involving generalized harmonic numbers are closely related to the central binomial coefficients, Catalan numbers and Bell polynomials. Among them, Bell polynomial is a combination polynomial. In this paper, inspired by Theorem 2.1 in paper  in the process of studying the relation between Bell polynomials and generalized harmonic numbers, we prove an integral expression about generalized harmonic numbers by combining initial conditions with double recursive relations. The integral identity reveals some relations between generalized harmonic numbers and Bell polynomials. At the same time, in view of the integral identity, we give the method of Log-integral transformation. And then making use of the transformation, we derive many harmonic number summation formulae with certain mathematical constants such as π, the Euler-Mascheroni constant γ, the Catalan constant G and the Apéry constant ζ (3).

References

[1] Liu, H. and Wang, W. (2019). Gauss's theorem and harmonic number summation formulae with certain mathematical constants. Journal of Difference Equations and Applications, 25(1/4), 313-330.

[2] Wang, X. and Chu, W. (2020). Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients. The Ramanujan Journal, 52, 641-668.

[3] Chen, H. (2022). Interesting Ramanujan-like series associated with powers of central binomial coefficients. Journal of Integer Sequences, 1(25).

[4] Boyadzhiev, K. N. (2011). Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities. Indian Journal of Pure and Applied Mathematics, 42, 371-386.

[5] Lehmer, D. H. (1985). Interesting Series Involving the Central Binomial Coefficient. The American Mathematical Monthly, 92(7), 449-457.

[6] Guillera, J. (2013). More hypergeometric identities related to Ramanujan-type series. The Ramanujan Journal, 32, 5-22.

How to cite this paper

Harmonic Number Identities from Log-integral Transformation

How to cite this paper:  Kunzhen Zhang, Xinhua Xiong. (2023) Harmonic Number Identities from Log-integral TransformationJournal of Applied Mathematics and Computation7(1), 83-89.

DOI: http://dx.doi.org/10.26855/jamc.2023.03.008