A characterization of the multivariate Gaussian distribution with a fixed mean and covariance matrix is that it is the unique probability distribution with fixed mean and covariance matrix that maximizes differential entropy. That is, the Gaussian PDF $f$ is the unique solution to the following maximization problem over probability densities:
$$\text{argmax} \{\mathbb{E}_f[\log(f(x))] : \mathbb{E}_f[x_i] = \mu_i,\; \mathbb{E}_f[x_i x_j] = \sigma_{ij}\}.$$
When there is a solution to this problem if we specify the moments of degree at most $k$ to the probability distribution? That is, when is there a solution to the problem
$$ \text{argmax} \{\mathbb{E}[\log(f(x))] : \forall \alpha,\; \mathbb{E}_f[x^{\alpha}] = \mu_{\alpha}\}. $$
Here, $\alpha = (\alpha_1, \dots, \alpha_n)$ ranges over those values where $\sum_{i=1}^n \alpha_i \le k$.
For example, it is obviously necessary that there exists a probability distribution with these given moments for there to be a solution (I've heard the question of whether or not there exists such a probability distribution is referred to as a moment problem in some contexts). Is this sufficient? Has this particular family of probability distributions been referred to in the literature before?
In the scalar case, the solution should have the form
$$f(x)= e^{- p(x)}$$ where $p(x)$ is a polynomial of degree $k$. The $k+1$ coefficients should be given implicitly by the $k$ moments (plus the normalization condition).
This suggests that the problem has no solution if $k$ is odd (as is the case for $k=1$).