  A new family of fractional counting processes based on a three-parameter
generalized Mittag-Leffler function was introduced and studied. As applications
we develop a fractional generalized compound process, introduce and develop
fractional generalized Bell polynomials and numbers, fractional generalized
Stirling numbers of the second kind, and a new family of quantum coherent
states. Stretched quantum coherent states, which are a generalization of the
famous Schr\"odinger-Glauber coherent states, were also introduced and studied.
In particular cases, the presented results reproduce known equations for
Poisson and fractional Poisson probability distributions, Bell numbers and
fractional Bell numbers, Stirling numbers and fractional Stirling numbers of
the second kind, as well as for known quantum coherent states.
