= Solution
The <binomial series> gives
$$
1-\alpha(1-s)^\beta=(1-\alpha)+\sum_{k\ge1}\alpha(-1)^{k+1}\binom\beta k\,s^k.
$$
Thus $p_0=1-\alpha>0$ and, for $k\ge1$,
$$
\boxed{p_k=\frac{\alpha\beta(1-\beta)(2-\beta)\cdots(k-1-\beta)}{k!}>0,}
$$
with the factors after $\beta$ omitted for $k=1$. The signs follow from $0<\beta<1$. As $s\uparrow1$, the series has nonnegative terms and its sum increases to $G(1)=1$, so <monotone convergence> gives $\sum_{k\ge0}p_k=1$. These are therefore genuine <probabilities> on the nonnegative integers, proving that $G$ is a <probability generating function>. Its <expected value> is infinite, since $G'(s)=\alpha\beta(1-s)^{\beta-1}\to\infty$; finiteness of the mean is not required for this <branching process>.
Write $G_n(s)=1-A_n(1-s)^{\beta^n}$. The composition formula gives $A_{n+1}=A_n\alpha^{\beta^n}$, with $A_0=1$. Consequently $A_n=\alpha^{1+\beta+\cdots+\beta^{n-1}}$, and summing the <geometric series> yields \b[the explicit generation generating function]
$$
\boxed{G_n(s)=1-\alpha^{(1-\beta^n)/(1-\beta)}(1-s)^{\beta^n},\quad n\ge0.}
$$
This is the <fractional-power offspring generating function> and its iterates.
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