q-Euler Numbers and Polynomials Associated with Basic Zeta Functions

TL;DR

This paper introduces a q-extension of Euler numbers linked to basic zeta functions, exploring their properties and identities through p-adic q-integration, thus extending classical number theory concepts.

Contribution

It presents a novel q-analogue of Euler numbers and investigates their identities using fermionic p-adic q-integration, connecting them to q-analogue zeta functions.

Findings

Defined the q-Euler numbers as interpolations of q-analogue Euler zeta functions

Derived identities of q-Euler numbers using p-adic q-integration

Extended classical Euler number properties to a q-analogue setting

Abstract

In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.