Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Mittag-Leffler functions and their generalizations appear in a large variety of problems in different areas. When we move from total differential equations to fractional equations Mittag-Leffler functions come in naturally. Fractional reaction-diffusion problems in physical sciences and general input-output models in other disciplines are some of the examples in this direction. Some basic properties of Mittag-Leffler functions are examined first. Then representations in terms of Mellin-Barnes integrals are given, which are shown to yield many known and new results directly and easily. The results are presented in terms of statistical densities so that they are directly applicable to statistical distribution theory and stochastic processes. Several pathways are examined of exponential and gamma densities going to Mittag-Leffler densities and then Mittag-Leffler densities going to Levy and Linnik densities. Then multivariable and matrix variable extensions of several results are given. Various results and representations given in this paper are directly applicable in many practical situations and are very suitable for further development of the theory. The material is presented in easily understandable formats, even for a beginner.