A new trigonometric identity with applications
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arXiv
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866929529549750272 |
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| author | Sun, Zhi-Wei Pan, Hao |
| author_facet | Sun, Zhi-Wei Pan, Hao |
| contents | In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1907_08118 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A new trigonometric identity with applications Sun, Zhi-Wei Pan, Hao Combinatorics Number Theory In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$. |
| title | A new trigonometric identity with applications |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/1907.08118 |