Euler Numbers and Polynomials Associated with Zeta Functions
For s∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζE(s)=2∑n=1∞((−1)n/ns), and ζE(s,x)=2∑n=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...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2008/581582 |
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| author | Taekyun Kim |
| author_facet | Taekyun Kim |
| author_sort | Taekyun Kim |
| collection | DOAJ |
| description | For s∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζE(s)=2∑n=1∞((−1)n/ns), and ζE(s,x)=2∑n=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)=Ek∗, and ζE(−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. |
| format | Article |
| id | doaj-art-c7f8ec10c3e24152b06ec64c83110307 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-c7f8ec10c3e24152b06ec64c831103072025-02-03T01:29:59ZengWileyAbstract and Applied Analysis1085-33751687-04092008-01-01200810.1155/2008/581582581582Euler Numbers and Polynomials Associated with Zeta FunctionsTaekyun Kim0Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South KoreaFor s∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζE(s)=2∑n=1∞((−1)n/ns), and ζE(s,x)=2∑n=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)=Ek∗, and ζE(−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.http://dx.doi.org/10.1155/2008/581582 |
| spellingShingle | Taekyun Kim Euler Numbers and Polynomials Associated with Zeta Functions Abstract and Applied Analysis |
| title | Euler Numbers and Polynomials Associated with Zeta Functions |
| title_full | Euler Numbers and Polynomials Associated with Zeta Functions |
| title_fullStr | Euler Numbers and Polynomials Associated with Zeta Functions |
| title_full_unstemmed | Euler Numbers and Polynomials Associated with Zeta Functions |
| title_short | Euler Numbers and Polynomials Associated with Zeta Functions |
| title_sort | euler numbers and polynomials associated with zeta functions |
| url | http://dx.doi.org/10.1155/2008/581582 |
| work_keys_str_mv | AT taekyunkim eulernumbersandpolynomialsassociatedwithzetafunctions |