Definition:Falling Factorial

Definition

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

Let $k$ be a positive integer.

Then $x$ to the (power of) $k$ falling is:

$\displaystyle x^{\underline k} := \prod_{j=0}^{k-1} \left({x - j}\right) = x \left({x - 1}\right) \cdots \left({x - k + 1}\right)$

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

For other values of $k$, this formula may be used:

$\displaystyle x^{\underline k} = \frac {x!} {\left({x - k}\right)!} = \frac {\Gamma \left({x+1}\right)} {\Gamma \left({x - k + 1}\right)!}$

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.

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.5$