Definition:Rising Factorial

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

Let $$k$$ be a positive integer.

Then $$x$$ to the (power of) $$k$$ rising is:
 * $$x^{\overline k} \ \stackrel {\mathbf {def}} {=\!=} \ \prod_{j=0}^{k-1} \left({x + j}\right) = x \left({x + 1}\right) \cdots \left({x + k - 1}\right)$$

This is called the $$k$$th rising factorial power of $$x$$.

For other values of $$k$$, this formula may be used:
 * $$x^{\overline k} = \frac {\Gamma \left({x + k}\right)}{\Gamma \left({x}\right)}$$

where $$\Gamma$$ signifies the Gamma function.

It is clear from the definition of the factorial that:
 * $$1^{\overline k} = k!$$
 * $$k^{\overline 1} = k$$

Also:
 * $$x^{\overline k} = \left({x + k - 1}\right)^{\underline k}$$

where $$x^{\underline k}$$ is the $k$th falling factorial power of $x$.

Also See

 * Falling Factorial
 * Factorial
 * Gamma Function

Note on Notation
The notation $$x^{\overline k}$$ is due to Alfredo Capelli, who used it in 1893.

An alternative and more commonly seen version (though arguably not as good) is $$x^{\left({k}\right)}$$.

This is known as the Pochhammer function or (together with $$\left({x}\right)_k$$ for its falling counterpart) the Pochhammer symbol (after Leo August Pochhammer).

However, depending on the context, either $$\left({x}\right)_k$$ or $$x^{\left({k}\right)}$$ can be used to indicate the rising factorial. In the field of combinatorics $$x^{\left({k}\right)}$$ tends to be used, while in that of special functions you tend to see $$\left({x}\right)_k$$. Therefore the more intuitively obvious $$x^{\overline k}$$ is becoming the preferred symbol for this.