# 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: $(18)$