Consider a model similar to the simple branching process except that individuals in the nth generation reproduce according to a reproduction law {$p_{nk}, k \geq 0$} with generating function $\phi_{n}(s)=\Sigma_{k}p_{nk}s^{k}$. Let $Z_{n}$ be the number in the nth generation.
a) Express the generating function
$f_{n}(s)=Es^{Z_{n}}$
in terms of $\phi_{k}(s), k\geq0$ where $\phi_0(s)=s$
b) Express $m_n=EZ_n$ in terms of $\mu_i, i\geq0$ where $\mu_i=\phi'_i(1)$
I don't really understand the problem. Is it that each generation splits into k offspring with probability $p_{nk}$, so the probability changes every generation?