Exact and explicit forms and combinatorial content of Levy stable distributions

We briefly recall the origin, history and some physical applications of Levy stable probability distributions. We report on recent findings of exact and explicit expressions for one-sided (0 < \alpha < 1) and two-sided (1 < \alpha \leq 2) Levy stable densities g_{\alpha}(x), of index \alpha for all \alpha = l/k, with k and l positive integers. We shall exemplify analytically and graphically several examples of known and infinite ensemble of new formulae for such distributions. We observe that in one-sided case (0 < l/k < 1) g_{l/k}(x) is a solution of the Stieltjes moment problem with negative moments being integer combinatorial sequences of factorial type. This last property, when seen as a conventional Stieltjes moment problem, can be solved with the use of inverse Mellin transform. In this way we derive an explicit formulae for g_{l/k}(x) in terms of Meijer G functions. The problem of non-uniqueness of so obtained solutions is discussed. Work done with Karol A. Penson : Phys. Rev. Lett, 2010, in press, http://arxiv.org/abs/1007.0193