Explicit Formulas involving q-Euler Numbers and Polynomials

TL;DR

This paper explores explicit formulas connecting q-Euler and q-Bernoulli numbers and polynomials, deriving relations through generating functions, derivatives, and p-adic q-integrals, advancing the theoretical understanding of these q-analogues.

Contribution

It introduces new relations between q-Euler and q-Bernoulli numbers using generating functions and p-adic q-integrals, providing novel theoretical insights.

Findings

Derived relations between q-Euler and q-Bernoulli numbers

Established formulas using generating functions and derivatives

Connected q-Euler numbers with q-Bernoulli numbers via p-adic q-integrals

Abstract

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.