Note on the generalization of the higher order q-Genocchi numbers and q-Euler numbers

TL;DR

This paper generalizes higher order q-Euler and q-Genocchi numbers and polynomials using multivariate fermionic p-adic integrals, introducing new interpolation functions and distribution relations for these q-extensions.

Contribution

It introduces a comprehensive generalization of higher order q-Euler and q-Genocchi numbers and polynomials, including (h, q)-extensions, with new interpolation and distribution formulas.

Findings

Derived interpolation functions for the generalized numbers and polynomials.

Established distribution relations for q-extensions of these polynomials.

Presented new (h, q)-extensions with meaningful properties.

Abstract

In this paper we present the generalization of the higher order q-Euler numbers and q-Genocchi numbers and w-Genocchi numbers and polynomials of high order using the multivariate fermionic p-adic integral on Zp. We have the interpolation functions of these numbers and polynomials. We obtain the distribution relations for q-extensions of w-Euler and w-Genocchi polynomials. We also have the interesting relation for q-extensions of these polynomials. We define (h, q)-extensions of w-Euler and w-Genocchi polynomials of high order. We have the interpolation functions for (h, q)-extensions of these polynomials. Moreover, we obtain some meaningful results of (h, q)-extensions of w-Euler and w-Genocchi polynomials.