Open Access

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

Journal of Inequalities and Applications20072008:371295

DOI: 10.1155/2008/371295

Received: 1 November 2007

Accepted: 22 December 2007

Published: 24 December 2007

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 . 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 q-adic, q-integral on .

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Authors’ Affiliations

(1)
The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University
(2)
Department of Mathematics and Computer Science, KonKuk University
(3)
Department of Mathematics, Hannam University

Copyright

© Taekyun Kim et al. 2008

This article is published under license to BioMed Central Ltd. 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.