$q$-Genocchi Numbers and Polynomials Associated with Fermionic $p$-Adic Invariant Integrals on ${\mathbb{Z}}_{p}$

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}^{(k)}(x)={2}^{k}k!\big(\begin{smallmatrix}n+k\\ \vspace{0pt}k\end{smallmatrix}\big){\sum{}}_{l=0}^{\infty{}}{\sum{}}_{{d}_{0}+{d}_{1}+\cdots{}+{d}_{k}=k-1,{d}_{i}\in{}\mathbb{N}}{(-1)}^{l}{(l+x)}^{n}, where ${G}_{n+k,q}^{(k)}(x)$ are the $q$-Genocchi polynomials of order $k$.