On Genocchi Numbers and Polynomials

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rationalnumbers, the complex number field, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks about q-extension, q is variously considered as an indeterminate, a complex, q ∈ C, or a p-adic number, q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp, then we assume |q − 1|p < 1. The ordinary Genocchi polynomials are defined as the generating function: