Let $X$, $Y$ and $Z$ are random variables with density $$ f_{X,Y,Z}(x,y,z) = \frac{1}{\sqrt{\pi} \Gamma(\frac{n-1}{2}) \Gamma(\frac{n}{2})2^n}(xy-z^2)^{\frac{n-1}{2} - 1} e^{-\frac{x}{2}} e^{-\frac{y}{2}} $$ for $x > 0, y > 0$ and $xy - z^2 > 0$.
I need to prove that $X$ and $Y$ are independent random variables.
I suppose that at least one of variables has gamma distribution or maybe both, but i don't know how to write density as a product of a part dependent on $x$ and a part dependent on $y$.
Please for some tips.
It suffices to show the $X,\,Y$ joint pdf $f_{X,\,Y}(x,\,y):=\int_{-\sqrt{xy}}^{-\sqrt{xy}}f_{X,\,Y,\,Z}(x,\,y,\,z)dz$ is of the form $g(x)h(y)$. (In fact we can take $g=h$, because $f$ is $x\leftrightarrow y$-symmetric.) The substitution $z=\sqrt{xy}\sin\theta,\,\theta\in\left[-\frac{\pi}{2},\,\frac{\pi}{2}\right]$ suffices for a proof:$$f_{X,\,Y}(x,\,y)=\frac{(xy)^{n/2-1}\exp-\frac{x+y}{2}}{\sqrt{\pi}\Gamma(\frac{n-1}{2})\Gamma(\frac{n}{2})2^n}\int_{-\pi/2}^{\pi/2}\cos^{n-2}\theta d\theta.$$That finishes the proof, but if you know a little about the Beta function you can simplify the above to show $g(x)=\frac{1}{\Gamma(\frac{n}{2})}x^{n/2-1}\exp-\frac{x}{2}$. As you conjectured, $X,\,Y$ are Gamma-distributed. Indeed they are IIDs with $k=\frac{n}{2},\,\theta=2$.