Asymptotic behavior of zeros of Bessel function derivatives
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912646057426944 |
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| author | Baricz, Árpád Kumar, Pranav Ponnusamy, Saminathan |
| author_facet | Baricz, Árpád Kumar, Pranav Ponnusamy, Saminathan |
| contents | We derive two distinct asymptotic expansions for the zeros $j_{ν,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_ν^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $ν$, for which we also establish an explicit error bound. The second addresses the case when $ν\to \infty$ with fixed $k$ and it involves the zeros of Airy functions and their derivatives. These results extend and refine the classical work of Wong, Lang, and Olver on the zeros of Bessel functions. In the course of obtaining our main results, we also generalize several auxiliary results, which in turn provide a broader framework for the study of zeros of special functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of zeros of Bessel function derivatives Baricz, Árpád Kumar, Pranav Ponnusamy, Saminathan Classical Analysis and ODEs 33C10, 34E05, 34E20 We derive two distinct asymptotic expansions for the zeros $j_{ν,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_ν^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $ν$, for which we also establish an explicit error bound. The second addresses the case when $ν\to \infty$ with fixed $k$ and it involves the zeros of Airy functions and their derivatives. These results extend and refine the classical work of Wong, Lang, and Olver on the zeros of Bessel functions. In the course of obtaining our main results, we also generalize several auxiliary results, which in turn provide a broader framework for the study of zeros of special functions. |
| title | Asymptotic behavior of zeros of Bessel function derivatives |
| topic | Classical Analysis and ODEs 33C10, 34E05, 34E20 |
| url | https://arxiv.org/abs/2510.12353 |