On the $q$-Genocchi numbers and polynomials with weight zero and their applications

TL;DR

This paper explores the properties of $q$-Genocchi numbers and polynomials with weight zero, revealing their relations via $p$-adic $q$-integrals, Bernstein polynomials, and connections to $p$-adic $ ext{log} ext{gamma}$ functions.

Contribution

It introduces new relations involving $q$-Genocchi numbers and polynomials with weight zero using $p$-adic integrals and Bernstein polynomials, and links them to $p$-adic $ ext{log} ext{gamma}$ functions.

Findings

Relations between $q$-Genocchi numbers and polynomials established.

Connections to $p$-adic $ ext{log} ext{gamma}$ functions demonstrated.

Application of Bernstein polynomials in this context.

Abstract

In this paper, the authors deal with the $q$-Genocchi numbers and polynomials with weight zero. They discover some interesting relations via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and familiar basis Bernstein polynomials. Finally, the authors show that the $p$-adic $\log$ gamma functions are associated with the $q$-Genocchi numbers and polynomials with weight zero.