A generalization of the Lupaș \(q\)-analogue of the Bernstein operator

Abstract

We introduce a Stancu type generalization of the Lupaș \(q\)-analogue of the Bernstein operator via the parameter \(\alpha\). The construction of our operator is based on the generalization of Gauss identity involving \(q\)-integers. We establish the convergence of our sequence of operators in the strong operator topology to the identity, estimating the rate of convergence by using the second order modulus of smoothness. For \(\alpha\) and \(q\) fixed, we study the limit operator of our sequence of operators taking into account the relationship between two consecutive terms of the constructed sequence of operators.