  A new fractional non-homogeneous counting process has been introduced and
developed using the Kilbas and Saigo three-parameter generalization of the
Mittag-Leffler function. The probability distribution function of this process
reproduces for certain set of the fractality parameters the famous Poisson and
fractional Poisson probability distributions as well as the probability
distribution function of a counting process displaying stretched exponential
interarrival times distribution. Applications of the developed fractional
non-homogeneous counting process cover fractional compound process, the
generalization of combinatorial polynomials and numbers, statistics of the
fractional non-homogeneous counting process and a new representation of the
Kilbas and Saigo function.
