Factorial moment generating function

In probability theory and statistics, the factorial moment generating function of the probability distribution of a real-valued random variable "X" is defined as 
for all complex numbers "t" for which this expected value exists. This is the case at least for all "t" on the unit circle formula_2, see characteristic function. If "X" is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then formula_3 is also called probability-generating function of "X" and formula_4 is well-defined at least for all "t" on the closed unit disk formula_5.
The factorial moment generating function generates the factorial moments of the probability distribution.
Provided formula_3 exists in a neighbourhood of "t" = 1, the "n"th factorial moment is given by 
where the Pochhammer symbol ("x")"n" is the falling factorial
Example.
Suppose "X" has a Poisson distribution with expected value λ, then its factorial moment generating function is
(use the definition of the exponential function) and thus we have