Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind
In this paper, we introduce the probabilistic Bernoulli numbers, Cauchy numbers, and Euler numbers of order α associated with the random variable Y, utilizing the generating function approach. Meanwhile, by employing important tools from combinatorial analysis, such as the partial Bell polynomials a...
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| Language: | English |
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Taylor & Francis Group
2025-12-01
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| Series: | Applied Mathematics in Science and Engineering |
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| Online Access: | https://www.tandfonline.com/doi/10.1080/27690911.2025.2485250 |
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| author | Aimin Xu |
| author_facet | Aimin Xu |
| author_sort | Aimin Xu |
| collection | DOAJ |
| description | In this paper, we introduce the probabilistic Bernoulli numbers, Cauchy numbers, and Euler numbers of order α associated with the random variable Y, utilizing the generating function approach. Meanwhile, by employing important tools from combinatorial analysis, such as the partial Bell polynomials and the Lagrange inversion formula, we provide computational formulas for these numbers in terms of the probabilistic Stirling numbers of the second kind. Furthermore, we introduce the probabilistic Stirling numbers of the first kind, and derive a computational formula in terms of the probabilistic Stirling numbers of the second kind, which can be seen as a probabilistic version of the Schlömilch formula. |
| format | Article |
| id | doaj-art-862a070d5cd34f9e82da38862f6dd0a2 |
| institution | OA Journals |
| issn | 2769-0911 |
| language | English |
| publishDate | 2025-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Applied Mathematics in Science and Engineering |
| spelling | doaj-art-862a070d5cd34f9e82da38862f6dd0a22025-08-20T01:50:45ZengTaylor & Francis GroupApplied Mathematics in Science and Engineering2769-09112025-12-0133110.1080/27690911.2025.2485250Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kindAimin Xu0Institute of Mathematics, Zhejiang Wanli University, Ningbo, People's Republic of ChinaIn this paper, we introduce the probabilistic Bernoulli numbers, Cauchy numbers, and Euler numbers of order α associated with the random variable Y, utilizing the generating function approach. Meanwhile, by employing important tools from combinatorial analysis, such as the partial Bell polynomials and the Lagrange inversion formula, we provide computational formulas for these numbers in terms of the probabilistic Stirling numbers of the second kind. Furthermore, we introduce the probabilistic Stirling numbers of the first kind, and derive a computational formula in terms of the probabilistic Stirling numbers of the second kind, which can be seen as a probabilistic version of the Schlömilch formula.https://www.tandfonline.com/doi/10.1080/27690911.2025.2485250Probabilistic Bernoulli numbersprobabilistic Cauchy numbersprobabilistic Euler numbershigher orderprobabilistic Stirling number11B68 |
| spellingShingle | Aimin Xu Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind Applied Mathematics in Science and Engineering Probabilistic Bernoulli numbers probabilistic Cauchy numbers probabilistic Euler numbers higher order probabilistic Stirling number 11B68 |
| title | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind |
| title_full | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind |
| title_fullStr | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind |
| title_full_unstemmed | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind |
| title_short | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind |
| title_sort | computing a family of probabilistic numbers in terms of probabilistic stirling numbers of the second kind |
| topic | Probabilistic Bernoulli numbers probabilistic Cauchy numbers probabilistic Euler numbers higher order probabilistic Stirling number 11B68 |
| url | https://www.tandfonline.com/doi/10.1080/27690911.2025.2485250 |
| work_keys_str_mv | AT aiminxu computingafamilyofprobabilisticnumbersintermsofprobabilisticstirlingnumbersofthesecondkind |