A complete asymptotic expansion for the semi-exponential operators related to x³

In the present paper, we study the asymptotic properties of the semi-exponential operator connected with p( x) =x³ p x = 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( x) =x³ p x = 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.