In this study, we discuss the approximation properties of Gamma operators Gα for absolutely continuous and locally bounded functions by using results of probability theory and some inequalities with the method of Bojanic-Cheng. And then, metric form Ωu (ξ, λ) is used with asymptotic formula combining to calculate an convergence rate asymptotically of Gamma operators Gα for the bounded functions locally and also analysis techniques are used with Bojanic-Khan-Cheng’s method to calculate an optimal convergence rate of Gamma operators Gα for the functions which are absolutely continuous. Lastly, the convergence of the operators to a specific function is illustrated using Maple software.