p-adic interpolating function associated with modified Dirichlet's type of twisted q-euler numbers and polynomials with weight alpha

TL;DR

This paper introduces a new p-adic interpolating function linked to modified Dirichlet's twisted q-Euler polynomials with weight alpha, expanding q-calculus and number theory through generating functions, Mellin transforms, and L-functions.

Contribution

It develops a novel p-adic interpolating function for modified twisted q-Euler polynomials, connecting them with Dirichlet's L-functions and zeta functions, and explores their properties and relations.

Findings

Derived combinatorial relations for q-Euler numbers and polynomials.

Established Witt's and distribution formulas for these polynomials.

Constructed p-adic twisted Euler q-L functions interpolating at negative integers.

Abstract

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. Furthermore we derive witt's type formula and Distribution formula (Multiplication theorem) for modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. In section three, by applying Mellin transformation we define q-analogue of modified twisted q-l-functions of Dirichlet's type and also we deduce that it can be written as modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. Moreover we will find a link between modified twisted Hurwitz-zeta function and q-analogue of modified twisted q-l-functions of Dirichlet's type which yields a deeper insight into the effectiveness of this type of generalizations. In addition we consider q-analogue of partial zeta function and we derive behavior of the modified q-Euler L-function at s = 0. In final section, we construct p-adic twisted Euler q-L function with weight alpha and interpolate Dirichlet's type of twisted q-Euler polynomials with weight alpha at negative integers. Our new generating function possess a number of interesting properties which we state in this paper.