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Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials
Open AccessArticle

Note on Type 2 Degenerate q-Bernoulli Polynomials

Department of Mathematics, Sogang University, Seoul 121-742, Korea
Kwangwoon Institute for Advanced Studies, Kwangwoon University, Seoul 139-701, Korea
Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 914;
Received: 17 June 2019 / Revised: 5 July 2019 / Accepted: 12 July 2019 / Published: 13 July 2019
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s q-Bernoulli numbers, identities concerning degenerate q-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. View Full-Text
Keywords: type 2 degenerate q-Bernoulli polynomials; p-adic q-integral type 2 degenerate q-Bernoulli polynomials; p-adic q-integral
MDPI and ACS Style

Kim, D.S.; Dolgy, D.V.; Kwon, J.; Kim, T. Note on Type 2 Degenerate q-Bernoulli Polynomials. Symmetry 2019, 11, 914.

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