On series identities involving $\binom{4k}k$ and harmonic numbers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912849915281408 |
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| author | Jiang, Bo Sun, Zhi-Wei |
| author_facet | Jiang, Bo Sun, Zhi-Wei |
| contents | The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example, we prove the identities $$\sum_{k=1}^\infty \frac{\binom{4k}{k}}{16^k}\left((22k^2-92k+11)H_{4k}-\frac{449k-275}{2}-\frac{85}{12k}\right)=-151-\frac{80}{3}\log{2}$$ and $$ \sum_{k=0}^\infty\frac{\binom{4k}{k}((11k^2+8k+1)(10H_{4k}-17H_{2k})+2k+18)}{(3k+1)(3k+2)16^k}=8\log2,$$ which were previously conjectured by Z.-W. Sun. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On series identities involving $\binom{4k}k$ and harmonic numbers Jiang, Bo Sun, Zhi-Wei Number Theory Combinatorics 11B65, 05A19, 39B22 The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example, we prove the identities $$\sum_{k=1}^\infty \frac{\binom{4k}{k}}{16^k}\left((22k^2-92k+11)H_{4k}-\frac{449k-275}{2}-\frac{85}{12k}\right)=-151-\frac{80}{3}\log{2}$$ and $$ \sum_{k=0}^\infty\frac{\binom{4k}{k}((11k^2+8k+1)(10H_{4k}-17H_{2k})+2k+18)}{(3k+1)(3k+2)16^k}=8\log2,$$ which were previously conjectured by Z.-W. Sun. |
| title | On series identities involving $\binom{4k}k$ and harmonic numbers |
| topic | Number Theory Combinatorics 11B65, 05A19, 39B22 |
| url | https://arxiv.org/abs/2601.18645 |