Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials
In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain <i>p</i>-adic invariant integrals on <inline-formula> <math display="inline"> <semantics> <msub...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/7/847 |
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| author | Dmitry V. Dolgy Dae San Kim Jongkyum Kwon Taekyun Kim |
| author_facet | Dmitry V. Dolgy Dae San Kim Jongkyum Kwon Taekyun Kim |
| author_sort | Dmitry V. Dolgy |
| collection | DOAJ |
| description | In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain <i>p</i>-adic invariant integrals on <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>. In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials. |
| first_indexed | 2024-04-14T05:10:23Z |
| format | Article |
| id | doaj.art-fabe0db71b3e4a5ea7a6b9fe43f26ddc |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-14T05:10:23Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-fabe0db71b3e4a5ea7a6b9fe43f26ddc2022-12-22T02:10:36ZengMDPI AGSymmetry2073-89942019-07-0111784710.3390/sym11070847sym11070847Some Identities of Ordinary and Degenerate Bernoulli Numbers and PolynomialsDmitry V. Dolgy0Dae San Kim1Jongkyum Kwon2Taekyun Kim3Hanrimwon, Kwangwoon University, Seoul 139-701, KoreaDepartment of Mathematics, Sogang University, Seoul 121-742, KoreaDepartment of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, KoreaDepartment of Mathematics, Kwangwoon University, Seoul 139-701, KoreaIn this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain <i>p</i>-adic invariant integrals on <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>. In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.https://www.mdpi.com/2073-8994/11/7/847Bernoulli polynomialsdegenerate Bernoulli polynomialsrandom variablesp-adic invariant integral on Zpinteger power sums polynomialsStirling polynomials of the second kinddegenerate Stirling polynomials of the second kind |
| spellingShingle | Dmitry V. Dolgy Dae San Kim Jongkyum Kwon Taekyun Kim Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials Symmetry Bernoulli polynomials degenerate Bernoulli polynomials random variables p-adic invariant integral on Zp integer power sums polynomials Stirling polynomials of the second kind degenerate Stirling polynomials of the second kind |
| title | Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials |
| title_full | Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials |
| title_fullStr | Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials |
| title_full_unstemmed | Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials |
| title_short | Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials |
| title_sort | some identities of ordinary and degenerate bernoulli numbers and polynomials |
| topic | Bernoulli polynomials degenerate Bernoulli polynomials random variables p-adic invariant integral on Zp integer power sums polynomials Stirling polynomials of the second kind degenerate Stirling polynomials of the second kind |
| url | https://www.mdpi.com/2073-8994/11/7/847 |
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