multifactorial of non-integer

I want to calculate 12.1!!!!!! , Just for curiosity. (One of my friend texted the term to me for some complex reason.)

I searched for multifactorial in terms of gamma function or equation, and found nothing if factorial goes more than double. I know the answer is somewhere between 12*6 and 13*7*1, but can I be more accurate than linearly interpolating those values?

(By the way, I spend an hour to learn thatt 12.1!!≠(12.1!)!. I wonder how huge the answer will be if it was 6 repetitive factorial, but not for now.)

• How do you define the multifactorial for non-integers ? – Peter May 20 '16 at 12:09
• Since the notation is ambiguous, I think you should first ask your friend what he means, if three consecutive applications of the double factorial (its alternative form) or six consecutive applications of the factorial (i.e. $\Gamma(x+1)$). Or something else entirely. By the way, $\log_{10}((12!)!)\approx 10^{9.6}$, so you might want to question your friend's "complicated reasons". – user228113 May 20 '16 at 12:18
• You should read more carefully: According to your Wikipedia page en.wikipedia.org/wiki/… it would be with $k=6, z=12.1$ $$k^{(z-1)/k}\frac{\Gamma(1+z/k)}{\Gamma(1+1/k)}\approx 60.24157$$ – gammatester May 20 '16 at 12:20
• @gammatester That's not true. The formula you used is only true for integer values. Mathworld (mathworld.wolfram.com/DoubleFactorial.html) gives the true double factorial formula for all complex numbers: $z!!=2^{(1+2z-\cos(\pi z))/4}\pi^{(\cos(\pi z)-1)/4}\Gamma(1+\frac{1}{2}z)$ which follows the relation $z!=z!!(z-1)!!$ for all complex numbers – Jacob May 20 '16 at 12:43