Series involving central binomial coefficients and higher-order harmonic numbers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910038668345344 |
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| author | Sun, Zhi-Wei Zhou, Yajun |
| author_facet | Sun, Zhi-Wei Zhou, Yajun |
| contents | We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where $ \mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $, $ G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $ ζ(3):= \sum_{n=1}^\infty\frac1{n^3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12091 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Series involving central binomial coefficients and higher-order harmonic numbers Sun, Zhi-Wei Zhou, Yajun Number Theory High Energy Physics - Theory Combinatorics 11B65, 11F67, 11M06, 05A19 We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where $ \mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $, $ G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $ ζ(3):= \sum_{n=1}^\infty\frac1{n^3}$. |
| title | Series involving central binomial coefficients and higher-order harmonic numbers |
| topic | Number Theory High Energy Physics - Theory Combinatorics 11B65, 11F67, 11M06, 05A19 |
| url | https://arxiv.org/abs/2602.12091 |