Raw moment recursion for a compound Poisson distribution ($m_k=\lambda {\sum }_{j=0}^{k-1}\left(\genfrac{}{}{0ex}{}{k-1}{j}\right){\mu }_{j+1}{m}_{k-1-j}$)

Write $m_k=\mathbb{E}{S}^k$, $m_0=1$, and ${\mu }_j=\mathbb{E}{X}^j$. Apply the Poisson size-bias identity with $h(s)=s^{k-1}$, and expand $(S+X{)}^{k-1}$ using the binomial theorem and independence. A finite $k$th severity moment suffices. In particular, the aggregate expected value, variance, and third central moment are $\lambda {\mu }_1$, $\lambda {\mu }_2$, and $\lambda {\mu }_3$, respectively; the latter two use raw severity moments.