I'm having trouble with starting and completing this question: $X$ is a Gamma-distributed random variable, $X\sim\Gamma(k,\theta)$. What is the (two-dimensional) moment generating function of $(X, \log X)$?
It turns out the answer is supposed to be $\psi_{x,logX} (t,u)=\Gamma(p+u)/\Gamma(p) * a^u/(1-at)^{p+u}$
The $2$-dimensional moment-generating function is given by:
$$ M_X(u,v) = \mathbb{E}[e^{uX+v\log X}] = \mathbb{E}[X^v e^{uX}]=\int_{0}^{+\infty}x^v e^{ux} \frac{1}{\Gamma(k)\,\theta^k} x^{k-1}\,e^{-\frac{x}{\theta}}\,dx $$ hence:
$$ M_X(u,v) = \frac{\Gamma(k+v)}{\theta^k\,\Gamma(k)\left(\frac{1}{\theta}-u\right)^{k+v}}$$ provided that $v>-k$ and $u<\frac{1}{\theta}$.