Definition:Falling Factorial

Definition
Let $x$ be a real number (but usually an integer).

Let $n$ be a positive integer.

Then $x$ to the (power of) $n$ falling is:
 * $\displaystyle x^{\underline n} := \prod_{j \mathop = 0}^{n - 1} \left({x - j}\right) = x \left({x - 1}\right) \cdots \left({x - n + 1}\right)$

Also known as
This is referred to as the $n$th falling factorial power of $x$.

It can also be referred to as the $n$th falling factorial of $x$.

Also see

 * Definition:Rising Factorial
 * Definition:Factorial
 * Definition:Gamma Function


 * Rising Factorial in terms of Falling Factorial: $x^{\overline n} = \left({x + n - 1}\right)^{\underline n}$


 * Falling Factorial as Quotient of Factorials: $x^{\underline n} = \dfrac {x!} {\left({x - n}\right)!} = \dfrac {\Gamma \left({x + 1}\right)} {\Gamma \left({x - n + 1}\right)}$


 * Integer to Power of Itself Falling is Factorial: $n^{\underline n} = n!$


 * Number to Power of One Falling is Itself: $x^{\underline 1} = x$
 * Number to Power of Zero Falling is One: $x^{\underline 0} = 1$