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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2504.07142 |
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| _version_ | 1866909573507448832 |
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| author | Kreinin, Alexander Marchenko, Andrey Vinogradov, Vladimir |
| author_facet | Kreinin, Alexander Marchenko, Andrey Vinogradov, Vladimir |
| contents | We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^β= 1 - e^{-xy}$, with $x>0$ and $ β> 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_β(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized Lambert function Kreinin, Alexander Marchenko, Andrey Vinogradov, Vladimir General Mathematics 60E05, 60F05, 60E07 We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^β= 1 - e^{-xy}$, with $x>0$ and $ β> 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_β(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable. |
| title | On generalized Lambert function |
| topic | General Mathematics 60E05, 60F05, 60E07 |
| url | https://arxiv.org/abs/2504.07142 |