We develop a white noise calculus for pure jump Lévy processes on Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is a novel construction of a distribution space. This space inherits many of the nice properties of the classical Schwartz space, but differs severely in its behaviour at zero. We apply Minlos' theorem to this space and get a white noise measure on this space which satisfies the first condition of analyticity and which is non-degenerate. Furthermore we obtain generalized Charlier polynomials for all pure jump Lévy processes. We introduce Kondratiev test function and distribution spaces, the S-transform and Wick product. We proceed to establish a differential calculus by using a transfer principle on Poisson spaces.