An expression of harmonic numbers using Mneimneh's binomial sum and the Pascal inversion formula
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2025
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| _version_ | 1866902077643423744 |
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| author | Pain, Jean Christophe |
| author_facet | Pain, Jean Christophe |
| contents | <p>Recently, Mneimneh published an interesting binomial sum involving harmonic numbers, generalizing an identity previously highlighted by Paule and Schneider. The new identity was derived using probabilistic analysis. Campbell proposed afterwards two different proofs, based respectively on the Zeilberger algorithm and on beta-tye integrals. In the present work, we obtain, using the Mneimneh identity combined with the Pascal inversion formula, two expressions of the harmonic numbers. The first one, which is a sum rule, involves the Lerch transcendent, and the second one, which is an explicit form, involves the Gauss hypergeometric series.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_14916208 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | An expression of harmonic numbers using Mneimneh's binomial sum and the Pascal inversion formula Pain, Jean Christophe Mathematics Number theory Harmonic numbers Lerch transcendent Hypergeometric functions <p>Recently, Mneimneh published an interesting binomial sum involving harmonic numbers, generalizing an identity previously highlighted by Paule and Schneider. The new identity was derived using probabilistic analysis. Campbell proposed afterwards two different proofs, based respectively on the Zeilberger algorithm and on beta-tye integrals. In the present work, we obtain, using the Mneimneh identity combined with the Pascal inversion formula, two expressions of the harmonic numbers. The first one, which is a sum rule, involves the Lerch transcendent, and the second one, which is an explicit form, involves the Gauss hypergeometric series.</p> |
| title | An expression of harmonic numbers using Mneimneh's binomial sum and the Pascal inversion formula |
| topic | Mathematics Number theory Harmonic numbers Lerch transcendent Hypergeometric functions |
| url | https://doi.org/10.5281/zenodo.14916208 |