 Fractional-power offspring generating function

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 $k\ge 1$, 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.