Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials
The purpose of this paper is to introduce and study type 2 degenerate <i>q</i>-Bernoulli polynomials and numbers by virtue of the bosonic <i>p</i>-adic <i>q</i>-integrals. The obtained results are, among other things, several expressions for those polynomials, ide...
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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/2073-8994/11/7/914 |
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| author | Dae San Kim Dmitry V. Dolgy Jongkyum Kwon Taekyun Kim |
| author_facet | Dae San Kim Dmitry V. Dolgy Jongkyum Kwon Taekyun Kim |
| author_sort | Dae San Kim |
| collection | DOAJ |
| description | The purpose of this paper is to introduce and study type 2 degenerate <i>q</i>-Bernoulli polynomials and numbers by virtue of the bosonic <i>p</i>-adic <i>q</i>-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s <i>q</i>-Bernoulli numbers, identities concerning degenerate <i>q</i>-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers. |
| first_indexed | 2024-04-11T21:56:36Z |
| format | Article |
| id | doaj.art-459a004d15d04807b9030186fb37dd8b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T21:56:36Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-459a004d15d04807b9030186fb37dd8b2022-12-22T04:01:05ZengMDPI AGSymmetry2073-89942019-07-0111791410.3390/sym11070914sym11070914Note on Type 2 Degenerate <i>q</i>-Bernoulli PolynomialsDae San Kim0Dmitry V. Dolgy1Jongkyum Kwon2Taekyun Kim3Department of Mathematics, Sogang University, Seoul 121-742, KoreaKwangwoon Institute for Advanced Studies, Kwangwoon University, Seoul 139-701, KoreaDepartment of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, KoreaDepartment of Mathematics, Kwangwoon University, Seoul 139-701, KoreaThe purpose of this paper is to introduce and study type 2 degenerate <i>q</i>-Bernoulli polynomials and numbers by virtue of the bosonic <i>p</i>-adic <i>q</i>-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s <i>q</i>-Bernoulli numbers, identities concerning degenerate <i>q</i>-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers.https://www.mdpi.com/2073-8994/11/7/914type 2 degenerate q-Bernoulli polynomialsp-adic q-integral |
| spellingShingle | Dae San Kim Dmitry V. Dolgy Jongkyum Kwon Taekyun Kim Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials Symmetry type 2 degenerate q-Bernoulli polynomials p-adic q-integral |
| title | Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
| title_full | Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
| title_fullStr | Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
| title_full_unstemmed | Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
| title_short | Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
| title_sort | note on type 2 degenerate i q i bernoulli polynomials |
| topic | type 2 degenerate q-Bernoulli polynomials p-adic q-integral |
| url | https://www.mdpi.com/2073-8994/11/7/914 |
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