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
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)