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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doaj-459a004d15d04807b9030186fb37dd8b2020-11-24T22:11:20ZengMDPI 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 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Dae San Kim Dmitry V. Dolgy Jongkyum Kwon Taekyun Kim |
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 |
author_facet |
Dae San Kim Dmitry V. Dolgy Jongkyum Kwon Taekyun Kim |
author_sort |
Dae San Kim |
title |
Note on Type 2 Degenerate <i>q</i>-Bernoulli Polynomials |
title_short |
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_sort |
note on type 2 degenerate <i>q</i>-bernoulli polynomials |
publisher |
MDPI AG |
series |
Symmetry |
issn |
2073-8994 |
publishDate |
2019-07-01 |
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. |
topic |
type 2 degenerate q-Bernoulli polynomials p-adic q-integral |
url |
https://www.mdpi.com/2073-8994/11/7/914 |
work_keys_str_mv |
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