Some Applications of the Fractional Poisson Probability Distribution

New physical and mathematical applications of recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states have been introduced and studied. Mathematical applications are related to the number theory. We have developed fractional generalization of the Bell polynomials, the Bell numbers, and the Stirling numbers of the second kind. The fractional Bell polynomials appearance is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. The fractional Stirling numbers of the second kind have been introduced and applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been found. In the limit case when the fractional Poisson distribution becomes the well-known Poisson probability distribution all of the above listed new developments and implementations turn into the well-known results of the quantum optics and the number theory. ∗E-mail address : [email protected] 1 PACS numbers: 05.10.Gg; 05.45.Df; 42.50.-p.