  We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy
stable probability distributions of index \alpha, 0< \alpha <1, of fundamental
importance in random systems, for anomalous diffusion and fractional kinetics.
We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty,
satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}),
p>0, for all \alpha = 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 \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.
