On the Twisted q-Euler numbers and polynomials associated with basic q-l-functions

Let p be a fixed odd positive integer. Throughout this paper Zp, Qp, C and Cp are respectively denoted as the ring of p-adic rational integers, the field of p-adic rational numbers, the complex numbers field and the completion of algebraic closure of Qp. The p-adic absolute value in Cp is normalized so that |p|p = 1/p. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp. If q ∈ C, one normally assumes 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, cf. [1, 2, 15, 20]. We use the notations as