Exact and explicit probability densities for one-sided Levy stable distributions

Abstract

We study functions g_α(x) which are one-sided, heavy-tailed Levy stable probability distributions of index α, 0< α<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_α(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_α(x) dx = exp(-p^α), p>0, for all α= l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of αin order to adapt g_α(x) to a given experimental situation.