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Research Article New Approach to q-Euler Numbers and Polynomials Taekyun Kim,1 Lee-Chae Jang,2 Young-Hee Kim,1 and Seog-Hoon Rim3 1

Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea Department of Mathematics and Computer Science, Konkuk University, Chungju 380-701, South Korea 3 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea 2

Correspondence should be addressed to Young-Hee Kim, [email protected] Received 11 January 2010; Accepted 14 March 2010 Academic Editor: Binggen Zhang Copyright q 2010 Taekyun Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a new construction of the q-extensions of Euler numbers and polynomials. We present new generating functions which are related to the q-Euler numbers and polynomials. We also consider the generalized q-Euler polynomials attached to Dirichlet’s character χ and have the generating functions of them. We obtain distribution relations for the q-Euler polynomials and have some identities involving q-Euler numbers and polynomials. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the qEuler polynomials at negative integers.

1. Introduction Let C be the complex number field. We assume that q ∈ C with |q| < 1 and that the q-number is defined by xq  1 − qx /1 − q in this paper. Recently, many mathematicians have studied for q-Euler and q-Bernoulli polynomials and numbers see 1–18. Specially, there are papers for the q-extensions of Euler polynomials and numbers approaching with two kinds of viewpoint among remarkable papers see 7, 10. It is known that the Euler polynomials are defined by 2/et  1ext  ∞ n n0 En xt /n!, for |t| < π, and En  En 0 are called the nth Euler numbers. The recurrence formula for the original Euler numbers En is as follows:

E0  1,

E  1n  En  0,

if n > 0

1.1

2

Advances in Difference Equations

see 7, 10. As for the q-extension of the recurrence formula for the Euler numbers, Kim 10 had the following recurrence formula:

∗  E0,q

2q 2

,

⎧ ⎨2q  n ∗ and qE∗  1  En,q  ⎩0 

if n  0, if n ≥ 1,

1.2

∗ with the usual convention of replacing E∗ n by En,q . Many researchers have made a wider and deeper study of the q-number up to recently see 1–18. In the field of number theory and mathematical physics, zeta functions and l-functions interpolating these numbers in negative integers have been studied by Cenkci and Can 3, Kim 4–12, and Ozden et al. 16–18. This research for q-Euler numbers seems to be motivated by Carlitz who had constructed the q-Bernoulli numbers and polynomials for the first time. In 1, 2, Carlitz considered the recurrence formulae for the q-extension of the Bernoulli numbers as follows:

B0,q  1,



k qB  1 − Bk,q 

⎧ ⎨1