Analytic Continuation of weighted q-Genocchi numbers and polynomials

TL;DR

This paper explores the analytic continuation of weighted q-Genocchi numbers and polynomials, introduces a new formula for their associated zeta function, and investigates the zeros' dynamics of these polynomials.

Contribution

It presents a novel formula for the weighted q-Genocchi-Zeta function and introduces the concept of zeros' dynamics of the analytically continued polynomials.

Findings

Derived a new formula for the weighted q-Genocchi-Zeta function.

Introduced the concept of zeros' dynamics for these polynomials.

Provided insights into the properties of the analytically continued functions.

Abstract

In the present paper, we analyse analytic continuation of weighted q-Genocchi numbers and polynomials. A novel formula for weighted q-Genocchi- Zeta function {\zeta}G,q (s | {\alpha}) in terms of nested series of {\zeta}G,q (n | {\alpha}) is derived. Moreover, we introduce a novel concept of dynamics of the zeros of analytically continued weighted q-Genocchi polynomials.