My teacher claims that, given all factorial moments $E((X)_r) = E(\prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable.
The first thing I found out is that the ordinary moments can be deduced from the factorial moments and vice-versa, meaning that this is the same as the moment problem only in the discrete, positive case.
Indeed, we can clearly get $E((X)_r)$ from the $E(X^i)$ by expanding the polynomial and using linearity, and the other way can be obtained through an induction: $E(X^i)=E((X)_{i+1}-\sum_{j\leq i-1}\alpha_jX^j)$.
I have not been able to go any further, I really feel like we need some extra hypothesis about the values taken by the variable other than the fact that they're countable and in $\mathbb{R}^+$.