  Fox's H-function provide a unified and elegant framework to tackle several
physical phenomena. We solve the space fractional diffusion equation on the
real line equipped with a delta distribution initial condition and identify the
corresponding H-function by studying the small $x$ expansion of the solution.
The asymptotic expansions near zero and infinity are expressed, for rational
values of the index $\alpha$, in terms of a finite series of generalized
hypergeometric functions. In $x$-space, the $\alpha=1$ stable law is also
derived by solving the anomalous diffusion equation with an appropriately
chosen infinitesimal generator for time translations. We propose a new
classification scheme of stable laws according to which a stable law is now
characterized by a generating probability density function. Knowing this
elementary probability density function and bearing in mind the infinitely
divisible property we can reconstruct the corresponding stable law. Finally,
using the asymptotic behavior of H-function in terms of hypergeometric
functions we can compute closed expressions for the probability density
functions depending on their parameters $\alpha, \beta, c, \tau $. Known cases
are then reproduced and new probability density functions are presented.
