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...

Full description

Bibliographic Details
Main Authors: Dae San Kim, Dmitry V. Dolgy, Jongkyum Kwon, Taekyun Kim
Format: Article
Language:English
Published: MDPI AG 2019-07-01
Series:Symmetry
Subjects:
Online Access:https://www.mdpi.com/2073-8994/11/7/914
id doaj-459a004d15d04807b9030186fb37dd8b
record_format Article
spelling 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&#8217;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&#8217;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 AT daesankim noteontype2degenerateiqibernoullipolynomials
AT dmitryvdolgy noteontype2degenerateiqibernoullipolynomials
AT jongkyumkwon noteontype2degenerateiqibernoullipolynomials
AT taekyunkim noteontype2degenerateiqibernoullipolynomials
_version_ 1725806217224257536