Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

Dae Kim,Dojin Kim,Taekyun Kim,Han Kim

doi:10.3390/sym11050613

Dae Kim, Dojin Kim + Show 2 more

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https://doi.org/10.3390/sym11050613

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Journal: Symmetry	Publication Date: May 2, 2019
Citations: 19	License type: CC BY 4.0

Affiliation: Pusan National University, Kwangwoon University

Abstract

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.

Highlights

We are going to review some known results
We obtain some identities of symmetry involving the type 2 Bernoulli polynomials, the type 2 Euler polynomials, power sums of odd positive integers and alternating power sums of odd positive integers which are derived from certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p
For the derivation of those identities, we introduced certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p, which have built-in symmetries

Summary

Introduction

We are going to review some known results. We first recall the definitions of Bernoulli and Euler polynomials together with their type 2 polynomials. We introduce the bosonic p-adic integrals and the fermionic p-adic integrals on Z p that we need for the derivation of an identity of symmetry. The type 2 Bernoulli polynomials are defined by generating function et Symmetry 2019, 11, 613; doi:10.3390/sym11050613. Bn = bn (0) are called type 2 Bernoulli numbers. When x = 0, En = En (0) are called the type 2 Euler numbers. We obtain some identities of symmetry involving the type 2 Bernoulli polynomials, the type 2 Euler polynomials, power sums of odd positive integers and alternating power sums of odd positive integers which are derived from certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p. Laplace distributions whose moments are closely related to the type 2 Bernoulli and Euler numbers. We note that the results here have applications in such diverse areas as combinatorics, probability, algebra and analysis (see [11,12,13])

Some Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

Further Remarks

Conclusions