Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind.
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| Title: | Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind. |
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| Authors: | Xu, Aimin1 (AUTHOR) xuaimin1009@hotmail.com |
| Source: | Applied Mathematics in Science & Engineering. Dec2025, Vol. 33 Issue 1, p1-17. 17p. |
| Subjects: | Probabilistic number theory, Bernoulli numbers, Lagrange problem, Euler's numbers, Combinatorics |
| Abstract: | 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. [ABSTRACT FROM AUTHOR] |
| Copyright of Applied Mathematics in Science & Engineering is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 190352383 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Xu%2C+Aimin%22">Xu, Aimin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> xuaimin1009@hotmail.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Applied+Mathematics+in+Science+%26+Engineering%22">Applied Mathematics in Science & Engineering</searchLink>. Dec2025, Vol. 33 Issue 1, p1-17. 17p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Probabilistic+number+theory%22">Probabilistic number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Bernoulli+numbers%22">Bernoulli numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Lagrange+problem%22">Lagrange problem</searchLink><br /><searchLink fieldCode="DE" term="%22Euler's+numbers%22">Euler's numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Combinatorics%22">Combinatorics</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Applied Mathematics in Science & Engineering is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1080/27690911.2025.2485250 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 17 StartPage: 1 Subjects: – SubjectFull: Probabilistic number theory Type: general – SubjectFull: Bernoulli numbers Type: general – SubjectFull: Lagrange problem Type: general – SubjectFull: Euler's numbers Type: general – SubjectFull: Combinatorics Type: general Titles: – TitleFull: Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Xu, Aimin IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 27690911 Numbering: – Type: volume Value: 33 – Type: issue Value: 1 Titles: – TitleFull: Applied Mathematics in Science & Engineering Type: main |
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