Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of Levy processes with no gaussian component whose Levy measure is of the type pi(dx) = e(gamma x)nu(e(x) - 1) dx, where nu is the density of the stable Levy measure and gamma is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero, L. Chaumont, Conditioned stable Levy processes and the Lamperti representation, J. Appl. Probab. 43 (2006) 967-983] as the underlying Levy processes in the Lamperti representation of conditioned stable Levy processes. In this paper, We Compute explicitly the law of these Levy 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. (C) 2008 Elsevier B.V. All rights reserved.