Definition:Rising Factorial

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

Let $n$ be a positive integer.

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

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

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

Also see

 * Definition:Falling Factorial
 * Definition:Factorial
 * Definition:Gamma Function


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


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


 * One to Integer Rising is Integer Factorial: $1^{\overline n} = n!$


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