A new type of Szász–Mirakjan operators based on q-integers

Abstract

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.