Identities on the Bernoulli and Genocchi Numbers and Polynomials

Seog-Hoon Rim,Joohee Jeong,Sun-Jung Lee

doi:10.1155/2012/184649

Abstract

We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.

Highlights

Let p be a fixed odd prime number
Throughout this paper Zp, Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp
Let N be the set of natural numbers and Z N ∪ {0}

Summary

Introduction

Let p be a fixed odd prime number. Throughout this paper Zp, Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. 1.3 where Gn are the nth ordinary Genocchi numbers see 8, 15. 1.4 where Gn x are called the nth Genocchi polynomials see 14, 15. By 1.3 and 1.4 , we get Witt’s formula for the nth Genocchi numbers and polynomials as follows: xndμ−1 x. By 1.12 and 1.13 , we get reflection symmetric formula for the Bernoulli polynomials as follows: Bn 1 − x −1 nBn x , B0 1, B 1 n − Bn δ1,n. We investigate some properties of the fermionic p-adic integrals on Zp. In this paper, we investigate some properties of the fermionic p-adic integrals on Zp By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics

Identities on the Bernoulli and Genocchi Numbers and Polynomials

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