Generating function for Gamma distribution expectation I encountered this formula in my assignment:
$$X\sim \Gamma(\alpha, \beta), 1\le k < \alpha$$
$$ E(X^{-k})=\frac{\beta^k}{\prod^k_{i=1}(\alpha-i)} $$
And I wonder what would happen if $k$ is greater than $\alpha$.

The expectation is infinite if $k>\alpha$. The expectation is
$$\int_0^\infty x^{-k} \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x)\,dx$$
If $k>\alpha$, the power of $x$ in the integrand, $\alpha-k-1$ is less than -1, so the integral diverges at 0.
