Higher order factorials.

Question

The function $y=x!$ can be drawn using the gamma functions, what could function would give higher order factorials like $y=x!!$ where $x!! = x\cdot(x-2)\cdot(x-4)\cdot\ldots\cdot5\cdot3\cdot1$. I have this: $$\prod\limits_{n=0}^{\lfloor x/k\rfloor-1}(x-kn)$$ Where k is the amount of !'s but obviously this is only valid for integer k, but I'm looking for something which would make would make sense and be continuous for non-integers.

Answer 1

For the odd case you could take

$$\Gamma_2(z) := 2^{(z-1)/2} \int_0^{\infty} t^{(z-1) /2} e^{-t} dt $$

For $z=1$ you have 1, and différentiation by parts you get

$$ \Gamma_2(z) = 2^{(z-1) /2} \frac{z-1}{2} \int_0^{\infty} t^{(z-3) /2} e^{-t}dt = (z-1) \Gamma_2(z-2) $$

So that by induction you get the right thing. In the even case you just get the semifactorial with a constant in front coming front $z=2$, i.e $\Gamma(1/2) $ which i think is something related to $\pi$ (maybe $\pi/4$). You can go around this problem by multiplying by an oscillating function :

$$ \left ( 1+ \cos( \pi z/2)^2 \alpha \right ) 2^{(z-1) /2} \Gamma((z-1) /2) $$

Where $ \alpha = \Gamma(1/2) -1$