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Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

1
Hanrimwon, Kwangwoon University, Seoul 139-701, Korea
2
Department of Mathematics, Sogang University, Seoul 121-742, Korea
3
Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea
4
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 847; https://doi.org/10.3390/sym11070847
Received: 28 May 2019 / Revised: 25 June 2019 / Accepted: 26 June 2019 / Published: 1 July 2019
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications Ⅱ)
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Abstract

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials. View Full-Text
Keywords: Bernoulli polynomials; degenerate Bernoulli polynomials; random variables; p-adic invariant integral on Zp; integer power sums polynomials; Stirling polynomials of the second kind; degenerate Stirling polynomials of the second kind Bernoulli polynomials; degenerate Bernoulli polynomials; random variables; p-adic invariant integral on Zp; integer power sums polynomials; Stirling polynomials of the second kind; degenerate Stirling polynomials of the second kind
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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Dolgy, D.V.; Kim, D.S.; Kwon, J.; Kim, T. Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials. Symmetry 2019, 11, 847.

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