Definition:Rising Factorial

Definition

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

Let $k$ be a positive integer.

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

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

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

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

$\displaystyle x^{\overline k} = \frac {\Gamma \left({x + k}\right)}{\Gamma \left({x}\right)}$

where $\Gamma$ signifies the Gamma function.

It is clear from the definition of the factorial that:

• $1^{\overline k} = k!$
• $k^{\overline 1} = k$

Also:

$x^{\overline k} = \left({x + k - 1}\right)^{\underline k}$

where $x^{\underline k}$ is the $k$th falling factorial power of $x$.

Also See

• Falling Factorial
• Factorial
• Gamma Function

Note on Notation

The notation $x^{\overline k}$ is due to Alfredo Capelli, who used it in 1893.

An alternative and more commonly seen version (though arguably not as good) is $x^{\left({k}\right)}$.

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

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

Sources

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