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 ≤ x < ∞, satisfying ∫_0^∞ 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≤ 4 and present many new exact solutions for k > 4, all expressed in

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... erms of known functions. This will allow a 'fine-tuning' of α in order to adapt g_α(x) to a given experimental situation.