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Fox's H-function provide a unified and elegant framework to tackle severalphysical phenomena. We solve the space fractional diffusion equation on thereal line equipped with a delta distribution initial condition and identify thecorresponding H-function by studying the small $x$ expansion of the solution.The asymptotic expansions near zero and infinity are expressed, for rationalvalues of the index $\alpha$, in terms of a finite series of generalizedhypergeometric functions. In $x$-space, the $\alpha=1$ stable law is alsoderived by solving the anomalous diffusion equation with an appropriatelychosen infinitesimal generator for time translations. We propose a newclassification scheme of stable laws according to which a stable law is nowcharacterized by a generating probability density function. Knowing thiselementary probability density function and bearing in mind the infinitelydivisible property we can reconstruct the corresponding stable law. Finally,using the asymptotic behavior of H-function in terms of hypergeometricfunctions we can compute closed expressions for the probability densityfunctions depending on their parameters $\alpha, \beta, c, \tau $. Known casesare then reproduced and new probability density functions are presented.

One dimensional stable probability density functions for rational index $\bf 0<\alpha \leq 2$