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

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 ${\mathbb{Z}}_p$, we construct $p$-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula: $G_{n+k,q}^{\left(k\right)}\left(x\right)=2^{k}k!\left(\begin{array}{c}n+k\\ k\end{array}\right){\sum }_{l=0}^{\infty }{\sum }_{d_0+d_1+\cdots +d_k=k-1,d_i\in \mathbb{N}}{\left(-1\right)}^l{\left(l+x\right)}^n$, where $G_{n+k,q}^{\left(k\right)}\left(x\right)$ are the $q$-Genocchi polynomials of order $k$.