Beta素数分布的卷积
Convolution of beta prime distribution
摘要：
本文建立了β素分布与其自身卷积的一些恒等式，涉及到β分布的平方根.这些恒等式的证明依赖于由第一类Appell级数得到的广义超几何级数的变换和${}_3F_2(1)$的Rupae关系。利用自可分解性论证，应用恒等式导出了具有加倍特征的汇合型超几何函数的导数的完全单调性。利用概率的方法，得到了抛物柱面函数和第二类汇合型超几何函数的Turán不等式的一个简单证明.详细讨论了密尔比的情况。
英文摘要：
We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for ${}_3F_2(1)$. Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill's ratio is discussed in detail.
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