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.
Received: 13 April 2019 / Revised: 24 April 2019 / Accepted: 28 April 2019 / Published: 2 May 2019
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
, 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.
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).
Share & Cite This Article
MDPI and ACS Style
Kim, D.S.; Kim, H.Y.; Kim, D.; Kim, T. Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials. Symmetry 2019, 11, 613.
Kim DS, Kim HY, Kim D, Kim T. Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials. Symmetry. 2019; 11(5):613.
Kim, Dae S.; Kim, Han Y.; Kim, Dojin; Kim, Taekyun. 2019. "Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials." Symmetry 11, no. 5: 613.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.