We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_{ \alpha}(x), 0 \leq x < \infty, 0 < \alpha < 1. We demonstrate that the knowledge of one such a distribution g_{ \alpha}(x) suffices to obtain exactly g_{ \alpha^{ p}}(x), p=2, 3,... Similarly, from known g_{ \alpha}(x) and g_{ \beta}(x), 0 < \alpha, \beta < 1, we obtain g_{ \alpha \beta}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For \alpha rational, \alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g_{ l/k}(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration.
Categories: math-ph cond-mat.stat-mech math.MP