Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type π ( d x ) = e γ x ν ( e x − 1 ) d x , where ν is the density of the stable Lévy measure and γ is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab. 43 (2006) 967–983] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.