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

Source: [Exact and explicit probability densities for one-sided Levy stable distributions](<https://www.sota2.com/research/paper/68fa29e71e209131ce838d93>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T13:12:58.265Z

Authors: K. A. Penson, K. Gorska

Published: 2010-07-01T15:22:00.000Z

## Abstract

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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1007.0193>)

[PDF](<https://arxiv.org/pdf/1007.0193.pdf>)