Journal of Inequalities and Applications 
Volume 2008 (2008), Article ID 371295, 9 pages
doi:10.1155/2008/371295
Research Article

Note on q-Extensions of Euler Numbers and Polynomials of Higher Order

Taekyun Kim,1 Lee-Chae Jang,2 and Cheon-Seoung Ryoo3

 1The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University, Taegu 702-701, South Korea
2Department of Mathematics and Computer Science, KonKuk University, Chungju 143-701, South Korea
3Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Received 1 November 2007; Accepted 22 December 2007

Recommended by Paolo Emilio Ricci

Abstract

In 2007, Ozden et al. constructed generating functions of higher-order twisted (h,q)-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral on ℤp. By applying their generating functions, they derived the complete sums of products of the twisted (h,q)-extension of Euler polynomials and numbers. In this paper, we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-type q-Euler zeta functions. Finally, we will derive the new formula for “sums of products of q-Euler numbers and polynomials” by using fermionic p-adic, q-integral on ℤp.