Euler Numbers and Polynomials Associated with Zeta Functions
Keyword(s):
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.
2011 ◽
Vol 54
(1)
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pp. 121-125
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1996 ◽
Vol 16
(4)
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pp. 805-819
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2009 ◽
Vol 215
(3)
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pp. 1185-1208
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2020 ◽
2019 ◽
Vol 2019
(1)
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2012 ◽
Vol 2012
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pp. 1-14
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