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Series of Bessel of first kind

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Post Sat May 13, 2017 9:37 am

Posts: 48

Let $\alpha_k $ be real parameters such that $\sum \limits_{k=1}^{m} \left| \alpha_{k} \right| \leq \frac{\pi}{2}$ . Then it holds that:

$$ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\prod_{k=1}^{m} \mathcal{J}_{0}(2 \alpha_{k}n) }{n^{2}} =\frac{\pi^{2}}{12} - \frac{1}{2}\sum_{k=1}^{m}\alpha_{k}^{2} $$

where $\mathcal{J}_0$ denotes the Bessel function of first kind .

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