Falling and rising factorials
Mathematical functions / From Wikipedia, the free encyclopedia
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In mathematics, the falling factorial (sometimes called the descending factorial,[1] falling sequential product, or lower factorial) is defined as the polynomial
The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial,[1] rising sequential product, or upper factorial) is defined as
The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers.[2]
The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where n is a non-negative integer. It may represent either the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to denote the binomial coefficient .[3]
In this article, the symbol is used to represent the falling factorial, and the symbol is used for the rising factorial. These conventions are used in combinatorics,[4] although Knuth's underline and overline notations and are increasingly popular.[2][5] In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol is used to represent the rising factorial.[6][7]
When is a positive integer, gives the number of n-permutations (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size to a set of size . The rising factorial gives the number of partitions of an -element set into ordered sequences (possibly empty).[lower-alpha 1]