The moment problem asks whether there exists a random variable $X$ with given moments. One way to do this: if $X$'s MGF converges about a neighborhood of $0$, we know the MGF uniquely characterizes $X$ in which case the pdf can be retrieved by applying the inverse laplace transform to its MGF.
My question is, if the odd moments are such that $\sum_{k\geq0} \frac{\mathbb{E}X^{(2k+1)k}}{k!} s^k$ converges around $0$, is knowing the odd moments of $X$ sufficient to uniquely characterize/retrieve $X$?
Edit: PhoemueX's comment notes if we take all odd moments zero, then the inversion is non-unique. But I don't see how to show non-uniqueness for any other circumstance or know if this is a degenercy.
Motivation: Consider if $X$ is a non-negative random variable such that the MGF of $X$ exists and converges about an open interval containing $0$. Then its possible to retrieve $X$ from moments $\mathbb{E}X^{nk}, k\in \mathbb{N}$, for fixed $n$. This is since the MGF $\mathbb{E}e^{X^n t}$ converges too, hence uniquely characterizes $X^n$, but since $X$ was positive this gives $X$. (Note if $n$ is even, the positive restriction was necessary).