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
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
Academic Editor: Paolo Emilio Ricci
Copyright © 2008 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.
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.