  The Fokker-Planck equation is considered, which is connected to the birth and
death process with immigration by the Poisson transform. The fractional
derivative in time variable is introduced into the Fokker-Planck equation. From
its solution (the probability density function), the generating function (GF)
for the corresponding probability distribution is derived. We consider the case
when the GF reduces to that of the negative binomial distribution (NBD), if the
fractional derivative is replaced to the ordinary one. Formulas of the
factorial moment and the $H_j$ moment are derived from the GF. The $H_j$ moment
derived from the GF of the NBD decreases monotonously as the rank j increases.
However, the $H_j$ moment derived in our approach oscillates, which is
contrasted with the case of the NBD. Calculated $H_j$ moments are compared with
those given from the data in $p\bar{p}$ collisions and in $e^+e^-$ collisions.
