Definition:Falling Factorial/Notation
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Notation for Falling Factorial
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.
Sources
- 1893: Alfredo Capelli: L'analisi algebrica e l'interpretazione fattoriale delle petenze (Giornale di Matematiche di Battaglini Vol. 31: pp. 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