Falling factorial power
In mathematics, the falling factorial (sometimes called the descending factorial,falling sequential product, or lower factorial) is defined:
The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial,rising sequential product, or upper factorial) is defined:
The value of each is taken to be 1 (an empty product) when n=0.
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. 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). Knuth uses the term factorial powers to comprise rising and falling factorials.
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