Journal of Applied Mathematics and Computation

ISSN Print: 2576-0645 Downloads: 87900 Total View: 1370151
Frequency: quarterly ISSN Online: 2576-0653 CODEN: JAMCEZ

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


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).


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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.