Levy stable two-sided distributions: exact and explicit densities for asymmetric case

Abstract

We study the one-dimensional Levy stable density distributions g(alpha, beta; x) for -infty < x < infty, for rational values of index alpha and the asymmetry parameter beta: alpha = l/k and beta = (l - 2r)/k, where l, k and r are positive integers such that 0 < l/k < 1 for 0 <= r <= l and 1 < l/k <= 2 for l-k <= r <= k. We treat both symmetric (beta = 0) and asymmetric (beta neq 0) cases. We furnish exact and explicit forms of g(alpha, beta; x) in terms of known functions for any admissible values of alpha and beta specified by a triple of integers k, l and r. We reproduce all the previously known exact results and we study analytically and graphically many new examples. We point out instances of experimental and statistical data that could be described by our solutions.