Euler Numbers and Polynomials Associated with Zeta Functions

Abstract

For $s\in{}\mathbb{C}$, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ${\zeta{}}_{E}(s)=2{\sum{}}_{n=1}^{\infty{}}({(-1)}^{n}/{n}^{s})$, and ${\zeta{}}_{E}(s,x)=2{\sum{}}_{n=0}^{\infty{}}({(-1)}^{n}/{(n+x)}^{s})$. 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, ${\zeta{}}_{E}(-k)={E}_{k}^{\ast}$, and ${\zeta{}}_{E}(-k,x)={E}_{k}^{\ast}(x)$. 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.