For anyone unfamiliar with multifactorial notation, I will give a quick rundown of it (at least, to the best of my understanding) for non-negative integer values of $n$:
$$n!=n(n-1)(n-2)(n-3)...(n-a), (n-a) > 0$$ $$n!!=n(n-2)(n-4)(n-6)...(n-a), 2 \geq (n-a) > 0$$ $$n!!!=n(n-3)(n-6)(n-9)...(n-a), 3 \geq (n-a) > 0$$
A more generalised expression can be given like so, where $k$ represents the number of factorial symbols:
$$n!^{k}=\left( \begin{cases} 1 & n=0\\ n & 1\leq n\leq k \\ n(n-k)(n-2k)(n-3k)...(n-a) & n>k \end{cases} \right), k \geq (n-a) > 0$$
Now, this is great for when you're working with (primarily) positive integers, but I'm curious how you'd go about extending the definition such that it will be valid for all real and complex numbers. While a general definition for any multifactorial would be amazing, I am primarily just looking for for the definition regarding the triple factorial.
I already know it's possible to do so for the double factorial; $z!!=2^{(1+2z-\cos(\pi z))/4}\pi^{(\cos(\pi z)-1)/4}\Gamma(z/2+1), z \in \mathbb{C}$, so it has me hopeful that it's also possible to define $z!!!$ in a similar such manner.
from Hot Weekly Questions - Mathematics Stack Exchange
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