Composite Fractional Time Evolutions of Cumulants in Fractional Generalized Birth Processes

A fractional generalized pure birth process is studied based on the master equation approach. The exact analytic solution of the generating function of the probability density for the process is given in terms of the Gauss hypergeometric function. The expressions of moments are also obtained in an explicit form. The effects of fixed and distributed initial conditions are also elucidated. The effect of memory is demonstrated quantitatively in terms of the mean, variance, and the Fano factor. It is also shown how to discriminate a fractional generalized birth process from the fractional Yule-Furry process. Further, the appearance of composite fractional time evolutions of cumulants is elucidated in conjunction with the fractional Poisson and the fractional Yule-furry processes.