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-E...

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a three-term recurrence relation for these dual polynomials are explicitly obtained. The completeness property of dual alternative q-Charlier polynomials is also esta...