Euler Numbers and Polynomials Associated with Zeta Functions

Abstract

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