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I studied double factorial which known for natural number $$ n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula for $z\in C$ $$z!!=2^{\frac{z}{2}} \Gamma\left(\frac{z}{2}+1 \right) \left(\frac{\pi}{2}\right)^{\frac{\cos(\pi z)-1}{4}} $$ I tried to find why it used this formula

I think we can write formula of double factorial for natural numbers by $$ z!!=2^{\frac{z}{2}} \Gamma\left(\frac{z}{2}+1 \right) \left(\frac{\pi}{2}\right)^{\frac{1}{2}f(z)}$$ where $$f(z) = \begin{cases} 0 &\mathrm{if} \space z \space \mathrm{is\space even} \\ -1 & \mathrm{if} \space z \space \mathrm{is\space odd} \end{cases}$$ which mean $$ f(z)=\frac{(-1)^z-1}{2}$$ but since $(-1)^z=\cos(\pi z)$ So I think it used that to define double factorial for non positive integers (which is correct for natural numbers).

but by the same way we can find that $$ z!!=2^{\frac{z}{2}} \Gamma\left(\frac{z}{2}+1 \right) \frac{2^{-\frac{f(z)}{2}}}{\Gamma\left(1+\frac{f(z)}{2}\right)}$$ where $f(z)$ is $0$ if $z$ is even and is $-1$ if its odd which mean $$ z!!=2^{\frac{z}{2}} \Gamma\left(\frac{z}{2}+1 \right) \frac{2^{\frac{1-\cos(\pi z)}{4}}}{\Gamma\left(\frac{\cos(\pi z)+3}{4}\right)}$$ the last formula is also correct for all natural numbers.

My Question is why math world chose that formula ?

is my reason is correct ? or there are another reason for choosing that formula?

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  • $\begingroup$ See also en.wikipedia.org/wiki/Double_factorial#Generalizations $\endgroup$
    – Martin R
    Commented 2 days ago
  • $\begingroup$ Start by expressing the double factorials of integers via half-shift gamma functions. $\endgroup$
    – Gary
    Commented 2 days ago
  • $\begingroup$ @Gary does that explain that formula ? show your work please . $\endgroup$
    – Faoler
    Commented 2 days ago

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