Abstract and Applied Analysis
Volume 2008 (2008), Article ID 232187, 8 pages
doi:10.1155/2008/232187
Research Article

q-Genocchi Numbers and Polynomials Associated with Fermionic p-Adic Invariant Integrals on ℤp

Leechae Jang1 and Taekyun Kim2

1Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea
2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

Received 3 April 2008; Accepted 22 April 2008

Academic Editor: Ferhan Atici

Copyright © 2008 Leechae Jang and Taekyun Kim. 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

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionic p-adic invariant integral on ℤp, we construct p-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula: Gn+k,q(k)(x)=2kk!(n+kk)∑l=0∞∑d0+d1+⋯+dk=k−1,di∈ℕ(−1)l(l+x)n, where Gn+k,q(k)(x) are the q-Genocchi polynomials of order k.