论文标题
在Feller，Pollard和Mittag-Leffler功能的完整单调性$e_α（-x）$
On Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function $E_α(-x)$
论文作者
论文摘要
Pollard使用轮廓积分来表明Mittag-Leffler函数是正函数的拉普拉斯变换，从而证明它完全是单调的。他还列举了Feller通过“概率理论方法”发现结果的个人交流。 Feller在发表的作品中使用了双变量分布的二维拉普拉斯变换来得出Pollard的结果。但是，尽管在Feller的起点和Pollard结果本身中发生了稳定的分布，但两种方法都可以描述为分析。我们采用贝叶斯概率方法，该方法将先前的分布分配给稳定分布的比例参数。我们将Feller的方法作为此类作业的特定实例。贝叶斯框架可以对Pollard结果进行概括。这导致了Mittag-Leffler函数的新颖积分表示以及由稳定密度多项式倾斜引起的变体。
Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone. He also cited personal communication by Feller of a discovery of the result by "methods of probability theory". In his published work, Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the Pollard result. But both approaches may be described as analytic, despite the occurrence of the stable distribution in Feller's starting point and in the Pollard result itself. We adopt a Bayesian probabilistic approach that assigns a prior distribution to the scale parameter of the stable distribution. We present Feller's method as a particular instance of such assignment. The Bayesian framework enables generalisation of the Pollard result. This leads to a novel integral representation of the Mittag-Leffler function as well as a variant arising from polynomial tilting of the stable density.
