New series involving binomial coefficients (III)
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910017127448576 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3π}2.$$ Motivated by Galois theory, we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type, and apply it to find 26 new irrational series identities. For example, we conjecture that \begin{align*}&\sum_{k=1}^\infty\frac{(32(91\sqrt{33}-523))^{k}}{k^3\binom{2k}k^2\binom{3k}k} \left((91\sqrt{33}+891)k-33\sqrt{33}-225\right) \\&\qquad=320\left(\frac{11}3\sqrt{33}L_{-11}(2)-27L_{-3}(2)\right), \end{align*} where $ L_{d}(2)=\sum_{k=1}^\infty\frac{(\frac{d}k)}{k^2}$ for any integer $d\equiv0,1\pmod4$ with $(\frac{d}k)$ the Kronecker symbol. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New series involving binomial coefficients (III) Sun, Zhi-Wei Number Theory 11B65, 11M06, 05A19, 11R11 We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3π}2.$$ Motivated by Galois theory, we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type, and apply it to find 26 new irrational series identities. For example, we conjecture that \begin{align*}&\sum_{k=1}^\infty\frac{(32(91\sqrt{33}-523))^{k}}{k^3\binom{2k}k^2\binom{3k}k} \left((91\sqrt{33}+891)k-33\sqrt{33}-225\right) \\&\qquad=320\left(\frac{11}3\sqrt{33}L_{-11}(2)-27L_{-3}(2)\right), \end{align*} where $ L_{d}(2)=\sum_{k=1}^\infty\frac{(\frac{d}k)}{k^2}$ for any integer $d\equiv0,1\pmod4$ with $(\frac{d}k)$ the Kronecker symbol. |
| title | New series involving binomial coefficients (III) |
| topic | Number Theory 11B65, 11M06, 05A19, 11R11 |
| url | https://arxiv.org/abs/2506.01870 |