Let's say I have a continuous random variable $0\le X<\infty$ and I know it has first raw moment $m_1=\alpha$, and subsequent moments are defined recursively as:
$$m_{n+1}=(1+\beta n)\alpha m_n$$
for some positive constants $\alpha$, $\beta$. The $n$th raw moment is thus:
$$m_n=\alpha^{n}\prod_{i=0}^{n-1}(1+\beta i)$$
What is the RV's distribution?