Properties
The rising and falling factorials can be used to express a binomial coefficient:
Thus many identities on binomial coefficients carry over to the falling and rising factorials.
A rising factorial can be expressed as a falling factorial that starts from the other end:
This is a special case of the fact that the rising and falling factorials are related as follows:
The rising and falling factorials are well defined in any ring, and therefore x can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
The rising factorial can be extended to real values of n using the Gamma function provided x and x + n are complex numbers that are not negative integers:
and so can the falling factorial:
If D denotes differentiation with respect to x, one has
Read more about this topic: Pochhammer Symbol
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