Evaluations of $ \sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series
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| Format: | Preprint |
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2024
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| _version_ | 1866910304815808512 |
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| author | Sun, Zhi-Wei Zhou, Yajun |
| author_facet | Sun, Zhi-Wei Zhou, Yajun |
| contents | We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Evaluations of $ \sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series Sun, Zhi-Wei Zhou, Yajun Combinatorics Number Theory 05A10, 11M32, 11B65, 33B15 We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$. |
| title | Evaluations of $ \sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series |
| topic | Combinatorics Number Theory 05A10, 11M32, 11B65, 33B15 |
| url | https://arxiv.org/abs/2401.12083 |