Euler Numbers and Polynomials Associated with Zeta Functions

Taekyun Kim
January 2008
Abstract & Applied Analysis;2008, p1
Academic Journal
For s ϵ ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζE(s) = 2Σn=1∞((-1)n / ns, and ζE(s, x) = 2Σn=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) = E*k, and ζE(-k, x) = E*k(k). 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.


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