Definition:Falling Factorial

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

Let $$k$$ be a positive integer.

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

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

For other values of $$k$$, this formula may be used:
 * $$x^{\underline k} = \frac {x!} {\left({x - k}\right)!} = \frac {\Gamma \left({x+1}\right)} {\Gamma \left({x - k + 1}\right)!}$$

where $$x!$$ is the (conventional) factorial sign and $$\Gamma$$ signifies the Gamma function.

It is clear from the definition of the factorial that $$k^{\underline k} = k!$$.

It is also clear from the definition of the falling factorial power that $$k^{\underline 1} = k$$.

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

where $$x^{\overline k}$$ is the $k$th rising factorial power of $x$.

Also See

 * Rising Factorial
 * Factorial
 * Gamma Function

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

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

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

See the note on notation in the Rising Factorial entry.