## Abstract and Applied Analysis

### On the $q$-Extension of Apostol-Euler Numbers and Polynomials

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 296159, 10 pages.

Dates
First available in Project Euclid: 10 February 2009

https://projecteuclid.org/euclid.aaa/1234298999

Digital Object Identifier
doi:10.1155/2008/296159

Mathematical Reviews number (MathSciNet)
MR2466221

Zentralblatt MATH identifier
1247.11028

#### Citation

Kim, Young-Hee; Kim, Wonjoo; Jang, Lee-Chae. On the $q$ -Extension of Apostol-Euler Numbers and Polynomials. Abstr. Appl. Anal. 2008 (2008), Article ID 296159, 10 pages. doi:10.1155/2008/296159. https://projecteuclid.org/euclid.aaa/1234298999

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