Given a sample $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$, and let $k \in \mathbb{Z}$ such that $n \ge \min(2, 2 - k)$. Show that an unbiased estimator for $\sigma^k$ can be given as follows. \begin{equation} \Sigma_k = (\frac{n-1}{2})^{k/2}\frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n+k-1}{2})} (\sum_{i=1}^nX_i)^k \end{equation}
Now I know that the sample mean ofcourse has a normal distribution, and looking around on the internet I can find expressions for the $k$-th moment which involve the gamma function. Since there doesn't seem to be a simple closed form I'm assuming I can't really prove this lack of bias by directly computing the expected value of the estimator. However I don't really know of any theorems of proving unbiased estimators without it. So is there a way of directly proving this expected value or how do I go about this?