The modified q-Euler numbers and polynomials

Let p be a fixed odd prime. Throughout this paper Zp, Qp, C and Cp will respectively, 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. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(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, one normally assumes |q| < 1. If q ∈ Cp, then we assume |q − 1|p < p , so that q = exp(x log q) for |x|p ≤ 1. The ordinary Euler numbers are defined by the generating function as follows: