  Physical and mathematical applications of fractional Poisson probability
distribution have been presented. As a physical application, a new family of
quantum coherent states has been introduced and studied. As mathematical
applications, we have discovered and developed the fractional generalization of
Bell polynomials, Bell numbers, and Stirling numbers. Appearance of fractional
Bell polynomials is natural if one evaluates the diagonal matrix element of the
evolution operator in the basis of newly introduced quantum coherent states.
Fractional Stirling numbers of the second kind have been applied to evaluate
skewness and kurtosis of the fractional Poisson probability distribution
function. A new representation of Bernoulli numbers in terms of fractional
Stirling numbers of the second kind has been obtained. A representation of
Schlafli polynomials in terms of fractional Stirling numbers of the second kind
has been found. A new representations of Mittag-Leffler function involving
fractional Bell polynomials and fractional Stirling numbers of the second kind
have been discovered. Fractional Stirling numbers of the first kind have been
introduced and studied. Two new polynomial sequences associated with fractional
Poisson probability distribution have been launched and explored. The
relationship between new polynomials and the orthogonal Charlier polynomials
has also been investigated. In the limit case when the fractional Poisson
probability distribution becomes the Poisson probability distribution, all of
the above listed developments and implementations turn into the well-known
results of quantum optics, the theory of combinatorial numbers and the theory
of orthogonal polynomials of discrete variable.
