Improve the approximation order of Bernstein type operators

In this study, we present a generalization of the well-known Bernstein operators based on an odd positive integer r denoted by K_(n,r) (f;x), first, we begin by studying the simultaneous approximation where we prove that the operator K_(n,r)^((s) ) (f;x) convergence to the function f^((s) ) (x) then we introduce and prove the Voronovskaja-type asymptotic formula when (r=3) giving us the order of approximation O(n^(-2) ) which is better than the order of the classical Bernstein operators O(n^(-1) ) followed by the error theorem and at the end, we give a numerical example to show the error of a test function and its first derivative taking different values of r.