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Concept
Falling and rising factorials
Summary
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, 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.
The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , 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 (x)_n with yet another meaning, namely to denote the binomial coefficient
In this article, the symbol (x)_n is used to represent the falling factorial, and the symbol x^(n) is used for the rising factorial. These conventions are used in combinatorics,
although Knuth's underline and overline notations and are increasingly popular.
In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol (x)_n is used to represent the rising factorial.
When x is a positive integer, (x)_n 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 n to a set of size x. The rising factorial x^(n) gives the number of partitions of an n-element set into x ordered sequences (possibly empty).
The first few falling factorials are as follows:
The first few rising factorials are as follows:
The coefficients that appear in the expansions are Stirling numbers of the first kind (see below).
When the variable x is a positive integer, the number (x)_n is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size x.
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