Abstract and Applied Analysis

Euler Numbers and Polynomials Associated with Zeta Functions

Taekyun Kim

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Abstract

For s , the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζ E ( s ) = 2 n = 1 ( ( 1 ) n / n s ) , and ζ E ( 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, ζ E ( k ) = E k , and ζ E ( k , x ) = E k ( 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.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 581582, 11 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969166

Digital Object Identifier
doi:10.1155/2008/581582

Mathematical Reviews number (MathSciNet)
MR2407279

Zentralblatt MATH identifier
1145.11019

Citation

Kim, Taekyun. Euler Numbers and Polynomials Associated with Zeta Functions. Abstr. Appl. Anal. 2008 (2008), Article ID 581582, 11 pages. doi:10.1155/2008/581582. https://projecteuclid.org/euclid.aaa/1220969166


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