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 n = 0. These symbols are collectively called
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 above. 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 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/overline notations are increasingly popular. In the theory of special functions (in particular the hypergeometric function) the Pochhammer symbol (x)n is used to represent the rising factorial.
A useful list of formulas for manipulating the rising factorial in this last notation is given in (Slater 1966, Appendix I).
When x is a positive integer, (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 xPn 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. Feller describes (x)n as "the number of ways to arrange n flags on x flagpoles".
Since the falling factorials are a basis for the polynomial ring, we can re-express the product of two of them as a linear combination of falling factorials:
The coefficients of the (x)m+n−k, called connection coefficients, have a combinatorial interpretation as the number of ways to identify (or glue together) k elements each from a set of size m and a set of size n.
We also have a connection formula for the ratio of two Pochhammer symbols given by
Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:
goes back to A. Capelli (1893) and L. Toscano (1939), respectively. Graham, Knuth and Patashnik propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively.
Other notations for the falling factorial include P(x, n), xPn, Px,n, or xPn. (See permutation and combination.)
An alternate notation for the rising factorial x(n) is the less common (x)+n. When the notation (x)+n is used for the rising factorial, the notation (x)–n is typically used for the ordinary falling factorial to avoid confusion.
A generalization of the falling factorial in which a function is evaluated on a descending arithmetic sequence of integers and the values are multiplied is:
where −h is the decrement and k is the number of factors. The corresponding generalization of the rising factorial is
This notation unifies the rising and falling factorials, which are [x]k/1 and [x]k/−1, respectively.
For any fixed arithmetic function and symbolic parameters , related generalized factorial products of the form
may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of in the expansions of and then by the next corresponding triangular recurrence relation:
These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the f-harmonic numbers, .