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Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

Department of Mathematics, Sogang University, Seoul 04107, Korea
Department of Mathematics, Kwangwoon University, Seoul 01897, Korea
Department of Mathematics, Pusan National University, Busan 46241, Korea
Author to whom correspondence should be addressed.
Symmetry 2019, 11(5), 613;
Received: 13 April 2019 / Revised: 24 April 2019 / Accepted: 28 April 2019 / Published: 2 May 2019
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications Ⅱ)
PDF [281 KB, uploaded 15 May 2019]


The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers. View Full-Text
Keywords: type 2 Bernoulli polynomials; type 2 Euler polynomials; identities of symmetry; Laplace distribution type 2 Bernoulli polynomials; type 2 Euler polynomials; identities of symmetry; Laplace distribution
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.; Kim, H.Y.; Kim, D.; Kim, T. Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials. Symmetry 2019, 11, 613.

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