Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind
In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of s...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-11-01
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| Series: | Demonstratio Mathematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/dema-2022-0166 |
| _version_ | 1811320173576060928 |
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| author | Chen Xue-Yan Wu Lan Lim Dongkyu Qi Feng |
| author_facet | Chen Xue-Yan Wu Lan Lim Dongkyu Qi Feng |
| author_sort | Chen Xue-Yan |
| collection | DOAJ |
| description | In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions. |
| first_indexed | 2024-04-13T12:55:32Z |
| format | Article |
| id | doaj.art-1eaf334989764ace9a9c3c4d9b184d52 |
| institution | Directory Open Access Journal |
| issn | 2391-4661 |
| language | English |
| last_indexed | 2024-04-13T12:55:32Z |
| publishDate | 2022-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Demonstratio Mathematica |
| spelling | doaj.art-1eaf334989764ace9a9c3c4d9b184d522022-12-22T02:46:05ZengDe GruyterDemonstratio Mathematica2391-46612022-11-0155182283010.1515/dema-2022-0166Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kindChen Xue-Yan0Wu Lan1Lim Dongkyu2Qi Feng3College of Engineering, Key Laboratory of Intelligent Manufacturing Technology, Inner Mongolia Minzu University, Tongliao 028000, Inner Mongolia, ChinaCollege of Engineering, Inner Mongolia Minzu University, Tongliao 028000, Inner Mongolia, ChinaDepartment of Mathematics Education, Andong National University, Andong 36729, Republic of KoreaInstitute of Mathematics, Henan Polytechnic University, Jiaozuo 454003, ChinaIn this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.https://doi.org/10.1515/dema-2022-0166bernoulli numberidentityproductcosecantcotangenthyperbolic cosecanthyperbolic cotangentseries expansioncentral factorial number of the second kindclosed-form formulaprimary 11b68secondary 11b7333b10 |
| spellingShingle | Chen Xue-Yan Wu Lan Lim Dongkyu Qi Feng Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind Demonstratio Mathematica bernoulli number identity product cosecant cotangent hyperbolic cosecant hyperbolic cotangent series expansion central factorial number of the second kind closed-form formula primary 11b68 secondary 11b73 33b10 |
| title | Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind |
| title_full | Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind |
| title_fullStr | Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind |
| title_full_unstemmed | Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind |
| title_short | Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind |
| title_sort | two identities and closed form formulas for the bernoulli numbers in terms of central factorial numbers of the second kind |
| topic | bernoulli number identity product cosecant cotangent hyperbolic cosecant hyperbolic cotangent series expansion central factorial number of the second kind closed-form formula primary 11b68 secondary 11b73 33b10 |
| url | https://doi.org/10.1515/dema-2022-0166 |
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