Special, conjugate and complete scale functions for spectrally negative Levy processes

Following recent developments in Hubalek and Kyprianou [28] the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Levy processes which are completely explicit. This will follow as a consequence of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Levy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed here.