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Article

Evaluation of Harmonic Number Series Involving the Binomial Coefficient C(3n,n) in the Denominator by Integration

by
Kunle Adegoke
1 and
Robert Frontczak
2,*
1
Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria
2
Independent Researcher, 72762 Reutlingen, Germany
*
Author to whom correspondence should be addressed.
AppliedMath 2025, 5(1), 31; https://doi.org/10.3390/appliedmath5010031
Submission received: 29 October 2024 / Revised: 4 February 2025 / Accepted: 25 February 2025 / Published: 19 March 2025
Versions Notes

Abstract

Two classes of series involving differences of harmonic numbers and the binomial coefficients C(3n,n) are evaluated in closed form. The classes under consideration are ∑k=0∞H3k+1−Hk(3k+1)3kkkmzkand∑k=0∞H2k−Hk(3k+1)3kkkmzk, where z is a complex number and m (a non-negative integer) is an additional parameter. The tool that will be applied is integration in combination with complex analysis and partial fraction decomposition. Several remarkable integral values and difficult series identities are stated as special cases of the main results.
Keywords: harmonic number; infinite series; integral harmonic number; infinite series; integral

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MDPI and ACS Style

Adegoke, K.; Frontczak, R. Evaluation of Harmonic Number Series Involving the Binomial Coefficient C(3n,n) in the Denominator by Integration. AppliedMath 2025, 5, 31. https://doi.org/10.3390/appliedmath5010031

AMA Style

Adegoke K, Frontczak R. Evaluation of Harmonic Number Series Involving the Binomial Coefficient C(3n,n) in the Denominator by Integration. AppliedMath. 2025; 5(1):31. https://doi.org/10.3390/appliedmath5010031

Chicago/Turabian Style

Adegoke, Kunle, and Robert Frontczak. 2025. "Evaluation of Harmonic Number Series Involving the Binomial Coefficient C(3n,n) in the Denominator by Integration" AppliedMath 5, no. 1: 31. https://doi.org/10.3390/appliedmath5010031

APA Style

Adegoke, K., & Frontczak, R. (2025). Evaluation of Harmonic Number Series Involving the Binomial Coefficient C(3n,n) in the Denominator by Integration. AppliedMath, 5(1), 31. https://doi.org/10.3390/appliedmath5010031

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