A Note on the Generalized q - Euler Numbers and Polynomials with Weak Weight α

Many mathematicians have studied Euler numbers and Euler polynomials( see [1-11]). Euler polynomials posses many interesting properties and arising in many areas of mathematics and physics. In this paper we introduce the generalized q-Euler numbers and polynomials with weak weight α. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of qextension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C one normally assume that |q| < 1. If q ∈ Cp, we normally assume that |q − 1|p < p− 1 p−1 so that q = exp(x log q) for |x|p ≤ 1. Let UD(Zp) be the space of uniformly differentiable function on Zp. For g ∈ UD(Zp) the fermionic p-adic invariant q-integral on Zp is defined by Kim as follows: