Levy稳定双边分布：非对称情形的精确和显式密度
Levy stable two-sided distributions: exact and explicit densities for asymmetric case
摘要：
我们研究了一维Levy稳定密度分布g（alpha，beta; x）对于-infty < x < infty，对于指数alpha和不对称参数beta的有理数值：α = 1/k和β =（1 - 2 r）/k，其中l、k和r是正整数，使得对于0 <= r <= l，0 < l/k < 1，对于l-k <= r <= k，1 < l/k <= 2。我们处理对称（β = 0）和不对称（β neq 0）的情况下。我们提供确切和明确的形式g（α，β; x）的已知函数的任何允许值的α和β指定的三元组的整数k，l和r。我们重现所有以前已知的确切结果，我们研究分析和图形许多新的例子。我们指出的实验和统计数据，可以描述我们的解决方案的情况下。
英文摘要：
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.
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