On the $q$-Extension of Apostol-Euler Numbers and Polynomials
Young-Hee Kim, Wonjoo Kim, and Lee-Chae Jang
Source: Abstr. Appl. Anal. Volume 2008 (2008), 10 pages.
Abstract
Recently, Choi et al. (2008) have studied the $q$-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order $n$ and multiple Hurwitz zeta function. In this paper, we define Apostol's type $q$-Euler numbers ${E}_{n,q,\xi }$ and $q$-Euler polynomials ${E}_{n,q,\xi }(x)$. We obtain the generating functions of ${E}_{n,q,\xi }$ and ${E}_{n,q,\xi }(x)$, respectively. We also have the distribution relation for Apostol's type $q$-Euler polynomials. Finally, we obtain $q$-zeta function associated with Apostol's type $q$-Euler numbers and Hurwitz's type $q$-zeta function associated with Apostol's type $q$-Euler polynomials for negative integers.
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Permanent link to this document: http://projecteuclid.org/euclid.aaa/1234298999
Digital Object Identifier: doi:10.1155/2008/296159
Mathematical Reviews number (MathSciNet):
MR2466221
Zentralblatt MATH identifier:
05534779
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