Let $X$ and $Y$ be random variables whose joint probability distribution function is given by $$f(x, y)=\frac{\sqrt{3}}{\pi} \exp \left\{-2\left(x^{2}+y^{2}-x y\right)\right\}, \quad x, y \in \mathbb{R}$$
Two new random variables are defined as $U=X+Y$ and $V=X-Y$. Find $E[U^h \ V^k]$ for some positive integers $h,k$.
In this problem I am also asked to find joint moment-generating function of $U,V$ and I got that as $$M_{UV}(s,t)=e^{s^2 / 2} \times e^{t^{2} / 6}$$
Now $$E[U^h \ V^k]=E[e^{h\ln U+k\ln V}]$$ is joint moment-generating function of logarithm of random variables $U,V$. However, I am unable to find their joint moment-generating function.