A Note on the q -Euler Numbers and Polynomials with Weak Weight α

Abstract

We construct a new type of q-Euler numbers and polynomials with weak weight $\alpha :E_{n,q}^{\left(\alpha \right)}$, $E_{n,q}^{\left(\alpha \right)}\left(x\right)$, respectively. Some interesting results and relationships are obtained. Also, we observe the behavior of roots of the q-Euler numbers $E_{n,q}^{\left(\alpha \right)}$ and polynomials $E_{n,q}^{\left(\alpha \right)}\left(x\right)$ with weak weight α. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Euler polynomials $E_{n,q}^{\left(\alpha \right)}$ with weak weight α.