A new family of q-analogue of Genocchi numbers and polynomials of higher order

TL;DR

This paper introduces a new family of higher-order q-Genocchi polynomials and numbers, deriving identities and a q-Hurwitz-Zeta function that interpolates these polynomials at negative integers, with applications in number theory and physics.

Contribution

It generalizes q-Genocchi polynomials of higher order and establishes new identities and a q-Hurwitz-Zeta function related to these polynomials.

Findings

Derived identities for q-Genocchi polynomials

Defined a q-Hurwitz-Zeta type function interpolating at negative integers

Connected the polynomials to applications in number theory and mathematical physics

Abstract

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by applying q-Mellin transformation to generating function of q-Genocchi polynomials of higher order and so we define q-Hurwitz-Zeta type function which interpolates of this polynomials at negative integers.