On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials of Higher Order

Let p be an odd prime number. For n ∈ Z N ∪ {0}, let Cpn {ζ | ζpn 1} be the cyclic group of order p, and let Tp limn→∞Cpn ⋃ n≥0 Cpn Cp∞ be the space of locally constant functions in the p-adic number fieldCp. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C, one normally assumes that |q| < 1. If q ∈ Cp, one normally assumes that |1 − q|p < 1. In this paper, we use the notation