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.

36 pages