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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)
PDF [255 KB, uploaded 15 July 2019]


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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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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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