Generalized scale functions for spectrally negative Lévy processes

For a spectrally negative L\'evy process, scale functions appear in the solution of two-sided exit problems, and particular relation with Laplace transform first time it exits closed interval. In this paper, we consider more general functionals, which can depend simultaneously on values process its supremum up to time. These quantities will be expressed terms generalized functions, defined using excursion theory. case functional does not supremum, these coincide ones found literature, therefore results work are an extension them.