Comments on the Properties of Mittag-Leffler Function

Abstract

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schr\"odinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Sch\"odinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.