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
The factorial moment generating function generates the factorial moments of the probability distribution. Provided
where the Pochhammer symbol (x)n is the falling factorial
(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising 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