= Fractional-power offspring generating function
{title2=$G(s)=1-\alpha(1-s)^\beta$}
For $0<\alpha,\beta<1$, this <probability generating function> has probabilities $p_0=1-\alpha$ and $p_k=\alpha\beta(1-\beta)\cdots(k-1-\beta)/k!$ for $k\ge1$, where the product after $\beta$ is empty for $k=1$. All probabilities are positive and sum to one, while the <expected value> is infinite. In a <Galton-Watson process> starting with one individual, the generation-$n$ <probability generating function> is $G_n(s)=1-\alpha^{(1-\beta^n)/(1-\beta)}(1-s)^{\beta^n}$. The <branching extinction probability> is $1-\alpha^{1/(1-\beta)}$, obtained by taking the limit of $G_n(0)$ and using that extinction by generation $n$ is an increasing sequence of <events>.
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