有理指标$\bf 0<α\leq 2$的一维稳定概率密度函数
One dimensional stable probability density functions for rational index $\bf 0<α\leq 2$
摘要：
Fox的H函数提供了一个统一而优雅的框架来处理几种物理现象。在实数线上求解具有δ分布初始条件的空间分数阶扩散方程，并通过研究解的小$x$展开式来确定相应的H函数.对于指数$α$的有理值，用广义超几何函数的有限级数表示了零和无穷大附近的渐近展开式。在$x$ -空间中，通过适当选择时间平移的无穷小生成元求解反常扩散方程，得到了$α=1$稳定律。我们提出了一个新的分类方案的稳定的法律，根据该稳定的法律，现在的特点是生成概率密度函数。知道了这个基本概率密度函数，并考虑到无穷可分的性质，我们可以重建相应的稳定律。最后，利用H-函数关于超几何函数的渐近性质，计算出依赖于参数$α, β, c, τ$的概率密度函数的封闭表达式.已知的情况下，然后再现和新的概率密度函数。
英文摘要：
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 $α$, in terms of a finite series of generalized hypergeometric functions. In $x$-space, the $α=1$ 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 $α, β, c, τ$. Known cases are then reproduced and new probability density functions are presented.
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