New series involving binomial coefficients (II)
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915779270672384 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | In this paper, we evaluate some series of the form $$\sum_{k=1}^\infty\frac{ak^2+bk+c}{k(3k-1)(3k-2)m^k\binom{4k}k}.$$ For example, we prove that $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2π$$ and $$\sum_{k=1}^\infty\frac{415k^2-343k+62}{k(3k-1)(3k-2)(-8)^k\binom{4k}k}=-3\log2.$$ We also pose many new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56π^3.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03086 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New series involving binomial coefficients (II) Sun, Zhi-Wei Number Theory Combinatorics 05A19, 11B65, 11A07, 11B68, 33B15 In this paper, we evaluate some series of the form $$\sum_{k=1}^\infty\frac{ak^2+bk+c}{k(3k-1)(3k-2)m^k\binom{4k}k}.$$ For example, we prove that $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2π$$ and $$\sum_{k=1}^\infty\frac{415k^2-343k+62}{k(3k-1)(3k-2)(-8)^k\binom{4k}k}=-3\log2.$$ We also pose many new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56π^3.$$ |
| title | New series involving binomial coefficients (II) |
| topic | Number Theory Combinatorics 05A19, 11B65, 11A07, 11B68, 33B15 |
| url | https://arxiv.org/abs/2307.03086 |