Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866910854792871936 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | In this paper, via the beta function we evaluate some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,π\ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}π.$$
We also establish the following efficient formula for computing $\log n$ with $1<n\le 85/4$: \begin{align*} &\sum_{k=0}^\infty\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\binom{4k}{2k}}\\ \ \ &=6n(n+1)(n-1)^3\log n-32n(n+1)^2(n^2-4n+1), \end{align*} where $$P(n):=n^6-58n^5+159n^4+52n^3+159n^2-58n+1.$$ In addition, we pose some conjectures on series whose summands involve $\binom{2k}k/(\binom{3k}k\binom{6k}{3k})\ (k\in\mathbb N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08275 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$ Sun, Zhi-Wei Number Theory Combinatorics 11B65, 05A19, 11A07, 33B15 In this paper, via the beta function we evaluate some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,π\ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}π.$$ We also establish the following efficient formula for computing $\log n$ with $1<n\le 85/4$: \begin{align*} &\sum_{k=0}^\infty\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\binom{4k}{2k}}\\ \ \ &=6n(n+1)(n-1)^3\log n-32n(n+1)^2(n^2-4n+1), \end{align*} where $$P(n):=n^6-58n^5+159n^4+52n^3+159n^2-58n+1.$$ In addition, we pose some conjectures on series whose summands involve $\binom{2k}k/(\binom{3k}k\binom{6k}{3k})\ (k\in\mathbb N)$. |
| title | Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$ |
| topic | Number Theory Combinatorics 11B65, 05A19, 11A07, 33B15 |
| url | https://arxiv.org/abs/2204.08275 |