A note on the modified q-Genocchi numbers and polynomials with weight (\alpha,\beta) and their interpolation function at negative integers

TL;DR

This paper introduces modified q-Genocchi numbers and polynomials with weights, explores their properties through generating functions, and establishes an interpolation formula linking them to a new q-Zeta function, enhancing understanding of their structure.

Contribution

It develops a new class of weighted q-Genocchi numbers and polynomials, derives their generating functions, and establishes an interpolation formula via a q-Zeta function, extending existing theory.

Findings

Derived new generating functions with interesting properties.

Established an interpolation formula connecting q-Genocchi numbers and polynomials.

Presented distribution and Witt's type formulas for the generalized q-Genocchi numbers.

Abstract

The purpose of this paper concerns to establish modified q-Genocchi numbers and polynomials with weight ({\alpha},{\beta}). In this paper we investigate special generalized q-Genocchi polynomials and we apply the method of generating function, which are exploited to derive further classes of q-Genocchi polynomials and develop q-Genocchi numbers and polynomials. By using the Laplace-Mellin transformation integral, we define q-Zeta function with weight ({\alpha},{\beta}) and by presenting a link between q-Zeta function with weight ({\alpha},{\beta}) and q-Genocchi numbers with weight ({\alpha},{\beta}) we obtain an interpolation formula for the q-Genocchi numbers and polynomials with weight ({\alpha},{\beta}). Also we derive distribution formula (Multiplication Theorem) and Witt's type formula for modified q-Genocchi numbers and polynomials with weight ({\alpha},{\beta}) which yield a deeper insight into the effectiveness of this type of generalizations for q=Genocchi numbers and polynomials. Our new generating function possess a number of interesting properties which we state in this paper.