Euler Numbers and Polynomials Associated with Zeta Functions

Abstract

Fors∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined byζE(s)=2∑n=1∞((−1)n/ns), andζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complexs-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is,ζE(−k)=Ek∗, andζE(−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.