Summing Sneddon-Bessel series explicitly
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913633143881728 |
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| author | Durán, Antonio J. Pérez, Mario Varona, Juan L. |
| author_facet | Durán, Antonio J. Pérez, Mario Varona, Juan L. |
| contents | We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_α(x j_{m,ν})J_β(y j_{m,ν})} {j_{m,ν}^{2n+α+β-2ν+2} J_{ν+1}(j_{m,ν})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer, $α,β,ν\in \mathbb{C}\setminus \{-1,-2,\dots \}$ with $2\operatorname{Re} ν< 2n+1 + \operatorname{Re} α+ \operatorname{Re} β$ and $\{j_{m,ν}\}_{m\geq 0}$ are the zeros of the Bessel function $J_ν$ of order $ν$. As an application we prove some extensions of the Kneser-Sommerfeld expansion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08709 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Summing Sneddon-Bessel series explicitly Durán, Antonio J. Pérez, Mario Varona, Juan L. Classical Analysis and ODEs 33C10, 33C20 We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_α(x j_{m,ν})J_β(y j_{m,ν})} {j_{m,ν}^{2n+α+β-2ν+2} J_{ν+1}(j_{m,ν})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer, $α,β,ν\in \mathbb{C}\setminus \{-1,-2,\dots \}$ with $2\operatorname{Re} ν< 2n+1 + \operatorname{Re} α+ \operatorname{Re} β$ and $\{j_{m,ν}\}_{m\geq 0}$ are the zeros of the Bessel function $J_ν$ of order $ν$. As an application we prove some extensions of the Kneser-Sommerfeld expansion. |
| title | Summing Sneddon-Bessel series explicitly |
| topic | Classical Analysis and ODEs 33C10, 33C20 |
| url | https://arxiv.org/abs/2207.08709 |