<p>In this paper, we study the local approximation properties of a new class of operators based on Mittag-Leffler functions depending on three parameters, also known as the Prabhakar function. These generalized operators extend previously known forms and reveal a rich analytical structure. A noteworthy connection between the constructed operators and the confluent hypergeometric function is also established. The main result of the paper is a pointwise complete asymptotic expansion for functions of exponential growth. To achieve this, we prove a localization theorem, which serves as the foundation for deriving the asymptotic behavior.</p>