Abstract and Applied Analysis
Volume 2008 (2008), Article ID 581582, 11 pages
doi:10.1155/2008/581582
Research Article
Euler Numbers and Polynomials Associated with Zeta Functions
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Received 16 January 2008; Accepted 19 April 2008
Academic Editor: Lance Littlejohn
Copyright © 2008 Taekyun Kim. 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.
Abstract
For s∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta
function are defined by ζE(s)=2∑n=1∞((−1)n/ns), and ζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complex s-plane, and these zeta functions have the values of the Euler numbers or the Euler
polynomials at negative integers. That is, ζE(−k)=Ek∗, and ζE(−k,x)=Ek∗(x). We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.