Falling and rising factorials  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 , where 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 formula_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, although Knuth's underline/overline notations formula_4 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 is used to represent the rising factorial. A useful list of formulas for manipulating the rising factorial in this last notation is given in . When is a positive integer, gives the number of -permutations of an -element set, or equivalently the number of injective functions from a set of size to a set of size , so is "the number of ways to arrange "n" flags on "x" flagpoles". In this context, other notations like and "P"("x", "n") are also sometimes used. The first few rising factorials are as follows: The first few falling factorials are as follows: The coefficients that appear in the expansions are Stirling numbers of the first kind. The rising and falling factorials are simply related to one another:  The rising and falling factorials are directly related to the ordinary factorial:  The rising and falling factorials can be used to express a binomial coefficient:  Thus many identities on binomial coefficients carry over to the falling and rising factorials. The rising and falling factorials are well defined in any unital ring, and therefore "x" can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.