Definition:Rising 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) $n$ rising is defined as:

$\displaystyle x^{\overline n} := \prod_{j \mathop = 0}^{n - 1} \paren {x + j} = x \paren {x + 1} \cdots \paren {x + n - 1}$


Also known as

This is referred to as the $n$th rising factorial power of $x$.

It can also be referred to as the $n$th rising factorial of $x$.


Notation

The notation $x^{\overline n}$ for $x$ to the $n$ rising 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 $x^{\left({n}\right)}$.

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


However, depending on the context, either $\left({x}\right)_n$ or $x^{\left({n}\right)}$ can be used to indicate the rising factorial. In the field of combinatorics $x^{\left({n}\right)}$ tends to be used, while in that of special functions you tend to see $\left({x}\right)_n$. Therefore the more intuitively obvious $x^{\overline n}$ is becoming the preferred symbol for this.


See the note on notation in the Falling Factorial entry.


Also see

  • Results about rising factorials can be found here.


Sources