Abstract
In this paper, we develop the unbounded extension of the Hille-Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Levy-Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grunwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.