Definition:Falling Factorial

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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 $x$.


Notation

The notation $x^{\underline n}$ for $x$ to the $n$ falling is due to Alfredo Capelli, who used it in $1893$.

This is the notation of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$.


A more commonly seen notation (though arguably not as good) is $\left({x}\right)_n$.

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


See the note on notation in the Rising Factorial entry.


Also see

  • Results about falling factorials can be found here.


Sources