In this paper we study that the q-Euler numbers and polynomials are analytically continued to Eq(s). A new formula for the Euler’s q-Zeta function ζE,q(s) in terms of nested series of ζE,q(n) is derived. Finally we introduce the new concept of the dynamics of analytically continued q-Euler numbers and polynomials. 2000 Mathematics Subject Classification 11B68, 11S40 Key wordsq-Bernoulli polynom...