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.
Best Answer
The expression mentioned in your question is a reflection of the fact that there are actually two "nice" extensions of the double factorial (one agreeing with it on even numbers and the other agreeing on odd numbers) and you can only combine them in a somewhat ad-hoc way.
Namely, the even and odd double factorials are
$$z!!_0 = \frac{2^{\frac{z}{2}} \Gamma\left(1+\frac{z}{2}\right)}{\Gamma\left(1+\frac{0}{2}\right)} = 2^{\frac{z}{2}} \Gamma\left(1+\frac{z}{2}\right)$$
and
$$z!!_1 = 1 \frac{2^{\frac{z-1}{2}} \Gamma\left(1+\frac{z}{2}\right)}{\Gamma\left(1+\frac{1}{2}\right)} = \sqrt{\frac{2}{\pi}} 2^{\frac{z}{2}} \Gamma\left(1+\frac{z}{2}\right).$$
The expression in your question is equal to
$$z!! = T_2(z) \: 2^{\frac{z}{2}} \Gamma\left(1+\frac{z}{2}\right),$$
where $T_2(z) = (\sqrt{2/\pi})^{(1-\cos (\pi z))/2}$ is equal to $1$ when $z$ is an even integer and to $\sqrt{2/\pi}$ when $z$ is an odd integer. I say that this is an ad-hoc construction because there are clearly many more possible $T_2(z)$ with this property (for example $(\sqrt{2/\pi})^{(1-\cos (13\pi z))/2}$).
Similarly, for the triple factorial we have three "nice" extensions, based on the triple factorials of numbers congruent to $0, 1, 2 \pmod 3$:
$$z!!!_0 = \frac{3^{\frac{z}{3}} \Gamma\left(1+\frac{z}{3}\right)}{\Gamma\left(1+\frac{0}{3}\right)},$$
$$z!!!_1 = \frac{3^{\frac{z-1}{3}} \Gamma\left(1+\frac{z}{3}\right)}{\Gamma\left(1+\frac{1}{3}\right)},$$
$$z!!!_2 = 2 \frac{3^{\frac{z-2}{3}} \Gamma\left(1+\frac{z}{3}\right)}{\Gamma\left(1+\frac{2}{3}\right)}.$$
So an extension of the triple factorial to $\mathbb{C}$ would be
$$z!!! = T_3(z) \: 3^{\frac{z}{3}} \Gamma\left(1+\frac{z}{3}\right),$$
where $T_3(z)$ is any function interpolating the three constants $1$, $3^{\frac{-1}{3}}/\Gamma\left(1+\frac{1}{3}\right)$ and $2 \cdot 3^{\frac{-2}{3}}/\Gamma\left(1+\frac{2}{3}\right)$ for $\mathbb{Z}\ni z=0,1,2 \pmod 3$ respectively, for example
$$T_3(z) = \left(\frac{3^{\frac{-1}{3}}}{\Gamma\left(1+\frac{1}{3}\right)} \right) ^{(1+2\cos(2\pi (z-1)/3))/3} \left(2 \frac{3^{\frac{-2}{3}}}{\Gamma\left(1+\frac{2}{3}\right)} \right)^{(1+2\cos(2\pi (z-2)/3))/3}.$$
You can see the resulting function here.
We can argue similarly for the higher-order multifactorials $z!^k$. The final expression is
$$z!^k = T_k(z) \: k^{\frac{z}{k}} \Gamma\left(1+\frac{z}{k}\right),$$
and a possible $T_k(z)$ generalizing the expressions for $T_2(z)$ and $T_3(z)$ given above is
$$T_k(z) = \prod_{j=1}^{k} \left(j\frac{k^{-j/k}}{\Gamma\left(1+\frac{j}{k}\right)} \right)^{E(k,j;z)},$$
where the exponent $E(k,j;z)=\frac{1}{k}\sum_{l=1}^k \cos\left(2\pi l \frac{(z-j)}{k}\right)$ is $1$ when $z = j \pmod k$ and vanishes when $z$ is any other integer. You can see its graph here (the slider also seems to allow fractional values of $k$, but I wouldn't trust that case).