Definition:Rising Factorial/Notation

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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.


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