Complete right tail asymptotic for the density of branching processes with fractional generating functions

The right tail asymptotic series consisting of attenuating exponential terms are derived for the densities of Galton-Watson processes with fractional probability generating functions. The frequencies in the exponential factors form fractal structures in the complex plane. We discuss conditions when the asymptotic series converges everywhere. The obtained right tail asymptotic is compared with the standard integral representation of the density and with the complete left tail asymptotic.