Some Identities on the q-Genocchi Polynomials of Higher-Order and q-Stirling Numbers by the Fermionic p-Adic Integral on ℤp

Let p be a fixed odd prime number. Throughout this paper, Zp,Qp,C, and Cp denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp. If q ∈ C, then one normally assumes |q| < 1. If q ∈ Cp, then we assume |q − 1|p < 1. In this paper, we use the following notation: