Eur. Phys. J. Special Topics 226, 3427–3443 (2017)
https://doi.org/10.1140/epjst/e2018-00073-1

Regular Article

Comments on the properties of Mittag-Leffler function

¹ ENEA – Frascati Research Center, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy
² H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul.Eliasza – Radzikowskiego 152, 31342 Krakow, Poland
³ Dep. of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

^a e-mail: silviakant@gmail.com

Received: 9 August 2017
Received in final form: 18 December 2017
Published online: 25 July 2018

Abstract

The properties of Mittag-Leffler function are 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 Schrödinger 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 Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.