In the present paper, we study the asymptotic properties of the semi-exponential operator connected with \(p\left( x\right) =x^{3}\) . The main result is a pointwise complete asymptotic expansion valid for locally smooth functions of exponential growth. All coefficients are derived and explicitly given. Furthermore, we characterize classes of functions f for which the semi-exponential operator connected with \(p\left( x\right) =x^{3}\) provides asymptotically a better approximation than the corresponding operator of exponential type. Finally, we present numerical examples which illustrate the better rate of convergence.
