q-Genocchi Numbers and Polynomials Associated with Fermionicp-Adic Invariant Integrals onℤp

Open Access

1 January 2008

journal article
research article
Published by Hindawi Limited in Abstract and Applied Analysis

Vol. 2008, 1-8
https://doi.org/10.1155/2008/232187

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

ℤ_{p}

, we construct

p

-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula:

G_{n + k, q}^{(k)} (x) = 2^{k} k! (\begin{smallmatrix} n + k \\ k \end{smallmatrix}) \sum_{l = 0}^{\infty} \sum_{d_{0} + d_{1} + \dots + d_{k} = k - 1, d_{i} \in ℕ} {(- 1)}^{l} {(l + x)}^{n}

, where

G_{n + k, q}^{(k)} (x)

are the

q

-Genocchi polynomials of order

k

Keywords

Funding Information

Kwangwoon University

This publication has 8 references indexed in Scilit:

10.1007/s11503-008-1006-9
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