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 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).[a]
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. For example, (8)3 = 8 × 7 × 6 = 336 is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. In this context, other notations like xPn, xPn, Pnx, or P(x, n) are also sometimes used. On the other hand, x(n) is "the number of ways to arrange n flags on x flagpoles",
where all flags must be used and each flagpole can have any number of flags. Equivalently, this is the number of ways to partition a set of size n (the flags) into x distinguishable parts (the poles), with a linear order on the elements assigned to each part (the order of the flags on a given pole).
The coefficients are called connection coefficients, and 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.
There is also a connection formula for the ratio of two rising factorials given by
Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:(p 52)
In this formula and in many other places, the falling factorial (x)n in the calculus of finite differences plays the role of xn in differential calculus. Note for instance the similarity of Δ (x)n = n (x)n−1 to d/d xxn = n xn−1 .
A similar result holds for the rising factorial and the backward difference operator.
goes back to A. Capelli (1893) and L. Toscano (1939), respectively. Graham, Knuth, and Patashnik(pp 47, 48)
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, Pnx, or xPn. (See permutation and combination.)
An alternative notation for the rising factorial x(n) is the less common (x)+ n. When (x)+ n is used to denote 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 x, t, 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 x in the expansions of (x)n,f,t 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,
Slater, Lucy J. (1966). Generalized Hypergeometric Functions. Cambridge University Press. Appendix I. MR0201688. — Gives a useful list of formulas for manipulating the rising factorial in (x)n notation.
Feller, William. An Introduction to Probability Theory and Its Applications. Vol. 1. Ch. 2.