Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_α(x), 0 \leq x < \infty, 0 < α< 1. We demonstrate that the knowledge of one such a distribution g_α(x) suffices to obtain exactly g_{α^{p}}(x), p=2, 3,... Similarly, from known g_α(x) and g_β(x), 0 < α, β< 1, we obtain g_{αβ}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For αrational, α= 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.

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