On generalized Lambert function

We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^{\beta }=1-e^{-xy}$, with $x>0$ and $\beta >1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_{\beta }(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable.