Explicit Formulas Involving q -Euler Numbers and Polynomials

Abstract

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 derive relations between the $q$-Euler numbers and $q$-Bernoulli numbers via the $p$-adic $q$-integral in the $p$-adic integer ring.