= Total progeny generating function of a branching process
{title2=$g(s)=sF(g(s))$}
For an almost surely finite <total progeny of a branching process>, let $F$ be the offspring <probability generating function> and $g$ that of the total, including one ancestor. Conditional on $k$ children, the total is one plus the sum of $k$ independent descendant totals. Independence and the <law of total expectation> give $g(s)=sF(g(s))$ for $|s|\leq1$, including the empty-product case $k=0$. If the offspring <expected value> is $m=F'(1-)<1$, differentiation for $0<s<1$ gives $g'(s)=F(g(s))/(1-sF'(g(s)))$. The denominator is positive because $F'(g(s))\leq m$. Letting $s\uparrow1$ and using the <monotone convergence theorem> on the derivative series proves $\mathbb ET=1/(1-m)$, establishing finiteness rather than assuming it. For <Poisson branching process> offspring, $F(z)=e^{m(z-1)}$.
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