Special, conjugate and complete scale functions for spectrally negative L\'evy processes

Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L'evy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative L'evy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.