# Definition:Rising Factorial/Notation

## Notation for Rising Factorial

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.

## Sources

- 1893: Alfredo Capelli:
*L'analisi algebrica e l'interpretazione fattoriale delle petenze*(*Giornale di Matematiche di Battaglini***Vol. 31**: 291 – 313) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials