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:
- $\ds x^{\underline n} := \prod_{j \mathop = 0}^{n - 1} \paren {x - j} = x \paren {x - 1} \cdots \paren {x - n + 1}$
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 $\paren x_n$.
This is known as the Pochhammer function or (together with $x^{\paren n}$ 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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(18)$