Pochhammer Symbol

In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation (x)_n, where n is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which interpretation is being used in any particular article. Pochhammer himself actually used (x)_n with yet another meaning, namely to denote the binomial coefficient .

In this article the Pochhammer symbol (x)_n is used to represent the falling factorial (sometimes called the "descending factorial", "falling sequential product", "lower factorial"):

In this article the symbol x(n) is used for the rising factorial (sometimes called the "Pochhammer function", "Pochhammer polynomial", "ascending factorial", "rising sequential product" or "upper factorial"):

These conventions are used in combinatorics (Olver 1999, p. 101). However in the theory of special functions (in particular the hypergeometric function) the Pochhammer symbol (x)_n is used to represent the rising factorial.

When x is a non-negative integer, then (x)_n gives the number of n-permutations of an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. However, for these meanings other notations like _xP_n and P(x,n) are commonly used. The Pochhammer symbol serves mostly for more algebraic uses, for instance when x is an indeterminate, in which case (x)_n designates a particular polynomial of degree n in x.