Is the Sibuya distribution a progeny?

Abstract

For $0<a<1$ the Sibuya distribution $s_a$ is concentrated on the set $\mathbb{N}^+$ of positive integers and is defined by the generating function $\sum_{n=1}^{\infty}s_a(n)z^n=1-(1-z)^a.$ A distribution $q$ on $\mathbb{N}^+$ is called a progeny if there exists a Galton-Watson process $(Z_n)_{n\geq 0}$ such that $Z_0=1$, such that $\mathbb{E}(Z_1)\leq 1$ and such that $q$ is the distribution of $\sum _{n=0}^{\infty}Z_n. $ The paper proves that $s_a$ is a progeny if and only if $\frac{1}{2}\leq a<1.$ The point is to find the values of $b=1/a$ such that the power series expansion of $u(1-(1-u)^b)^{-1}$ has non negative coefficients. The proof is not short, but elementary.