Probability distribution built by Prabhakar function. Related Tur\'an and Laguerre inequalities

Introducing the discrete probability distribution by means of the Prabhakar (or the three--parameter Mittag--Leffler) function, we establish explicit expressions for raw and factorial moments and also general fractional order moments. Applying an elementary moment inequality we obtain functional upper bounds for the Tur\'anian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.