Analytic Continuation of q-Euler numbers and polynomials

TL;DR

This paper extends q-Euler numbers and polynomials to a complex domain, introduces a new formula for their associated zeta function, and explores their dynamic properties.

Contribution

It provides the first analytic continuation of q-Euler numbers and polynomials and introduces a novel concept of their dynamics.

Findings

Derived a new formula for the q-Euler zeta function

Successfully analytically continued q-Euler numbers and polynomials

Introduced the concept of dynamics for these continued functions

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 $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials.