This paper investigates exponential-type operators generated by power functions of the form \(p(x) = x^\alpha \) , where \(\alpha \in \mathbb {R}\) . Building upon the framework introduced by May and further developed by Ismail and May, we derive explicit representations for these operators across various ranges of \(\alpha \) . While the cases for \(\alpha = 0, 1, 2, 3\) , and 3/2 were previously addressed in the literature, this work extends the theory by providing comprehensive formulations for \(\alpha > 2\) and remarks on the case \(\alpha < 1\) . For \(\alpha \in (1,2)\) , we recover the operators previously studied by Gupta. For \(\alpha > 2\) , we express the operators using Wright functions, showing their convergence properties for locally integrable functions. As a collateral result, we provide detailed proof for the computation of the inverse Laplace transform of the function \(\exp (-a\cdot s^b)\) . For \(\alpha < 1\) , we demonstrate that the operators can be represented using Fourier transforms, though they may lack positivity in some cases. Specific examples are provided for \(\alpha = 0\) (yielding the Gauss-Weierstrass operators), \(\alpha = 1/2\) (yielding operators based on Airy functions), and \(\alpha = 4m/(4m+1)\) where \(m \ge 1\) is an integer. These new families of operators offer interesting avenues for future research on approximation theory.
