Taylor coefficients and series involving harmonic numbers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917261236764672 |
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| author | Hou, Qing-Hu Sun, Zhi-Wei |
| author_facet | Hou, Qing-Hu Sun, Zhi-Wei |
| contents | During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{π^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 ζ(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0<j \le n}j^{-m}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03699 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Taylor coefficients and series involving harmonic numbers Hou, Qing-Hu Sun, Zhi-Wei Combinatorics Primary 05A19, 11B65, Secondary 33B15 During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{π^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 ζ(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0<j \le n}j^{-m}$. |
| title | Taylor coefficients and series involving harmonic numbers |
| topic | Combinatorics Primary 05A19, 11B65, Secondary 33B15 |
| url | https://arxiv.org/abs/2310.03699 |