A unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p adic q integral on Zp at q=-1

TL;DR

This paper unifies multiple twisted Euler and Genocchi numbers and polynomials using p-adic q-integrals, deriving new relations, formulas, and interpolation results that deepen understanding of their structure and generalizations.

Contribution

It introduces a unified framework for multiple twisted Euler and Genocchi numbers and polynomials via p-adic q-integrals, extending previous results and establishing new formulas and interpolation properties.

Findings

Derived relations between generating functions and p-adic q-integrals.

Established Witt's type formula for the unification.

Obtained a distribution (Multiplication) Theorem for the numbers and polynomials.

Abstract

The present paper deals with unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1. Some earlier results of Ozden's papers in terms of unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 can be deduced. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Euler polynomials and Genocchi polynomials. To be more precise we summarize our results as follows, we obtain some relations between H.Ozden's generating function and fermionic p-adic q-integral on Zp at q = 1. Furthermore we derive Witt's type formula for the unification of twisted Euler and Genocchi polynomials. Also we derive distribution formula (Multiplication Theorem) for multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 which yields a deeper insight into the effectiveness of this type of generalizations. Furthermore we define unification of multiple twisted zeta function and we obtain an interpolation formula between unification of multiple twisted zeta function and unification of the multiple twisted Euler and Genocchi numbers at negative integer. Our new generating function possess a number of interesting properties which we state in this paper.