TITLE: The double factorial notation
QUESTION [19 upvotes]: The double factorial is defined as
$$n!! = \begin{cases} n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1 = \dfrac{(n+1)!}{2^{(n+1)/2}((n+1)/2)!} & \text{ If $n \in \mathbb{Z}^+$, is odd}\\
n \cdot (n-2) \cdot (n-4) \cdots 4 \cdot 2 = 2^{n/2} (n/2)! & \text{ If $n \in \mathbb{Z}^+$, is even}\\
1 & \text{If $n \in \{0,-1\}$}
\end{cases} \,\,\,\,\,\,\,\,\,\,\,\,\,\, (\star)$$
I am curious about the origin of this notation. I see it to be a poor, non-suggestive notation. After a quick search, I see that the notation was first used in Arfken $1985$. I am curious as to why one would use such a non-suggestive notation, call it double factorial and more importantly, why have others caught on to this notation? Also, are there alternate notations to denote $(\star)$?

REPLY [4 votes]: Well, the way I assume it was chosen was:

With one factorial sign, we decrement by one:
$n! = n \cdot (n - 1) \cdot ... \cdot 2 \cdot 1$
So, if we have two factorial signs, let's decrement by two!
$n!! = n \cdot (n - 2) \cdot ... \cdot 3 \cdot 1$

And thus, it was done.
It's a terrible abuse of notation, as mentioned in the other answer, I'll agree.