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 fermionicp-adic invariant integral onℤp, we constructp-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, whereGn+k,q(k)(x)are theq-Genocchi polynomials of orderk.
Highlights
Let p be a fixed odd prime number
When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp
If q ∈ Cp, we assume that |1 − q|p < 1, see 1–6
Summary
The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. By using the fermionic p-adic invariant integral on Zp, we construct p-adic Genocchi numbers and polynomials of higher order. We derive the following interesting formula: Gnk k,q x. Gnk k,q x are the q-Genocchi polynomials of order k.
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