  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.
