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

We construct a new type of q-Euler numbers and polynomials with weak weight α : E α n,q , E α n,q x , respectively. Some interesting results and relationships are obtained. Also, we observe the behavior of roots of the q-Euler numbers E α n,q and polynomials E α n,q x with weak weight α. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Euler polynomials E α n,q with weak weight α.