Analytic Continuation of q-Euler numbers and polynomials

Abstract

In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\hat{I}{\P }_{E,q}(s)$ in terms of nested series of $\hat{I}{\P }_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials.

5 pages