β-伽马产品和信号规则
Produit Beta-Gamma et régularité du signe
摘要：
本文研究了两个随机变量$B(a,b)$和$Γ(c).$的独立乘积的乘性卷积核T的全正性。当$b$或$d = a+b -c$为整数时，该核是无限阶全正的。否则，T的符号正则性总是有限阶的，这是在这里计算的。更确切地说，对于每个$n\ge 1$它表明，T是全积极的$n + 1$阶当且仅当$(d,b)$位于一定的楼梯${\mathcal E}_n$绘制在上半平面。这个阶梯还刻画了与第二类汇合型超几何函数相关的几个行列式的符号不变性。
英文摘要：
We study the total positivity of the multiplicative convolution kernel T associated with the independent product of two random variables $B(a,b)$ and $Γ(c).$ This kernel is totally positive of infinite order if $b$ or $d = a+b -c$ are integers. Otherwise the sign-regularity of T has always a finite order, which is here computed. More precisely, for every $n\ge 1$ it is shown that T is totally positive of order $n + 1$ if and only if $(d,b)$ lies above a certain stairway ${\mathcal E}_n$ plotted in the upper half-plane. This stairway also characterizes the sign-invariance of several determinants associated with the confluent hypergeometric function of the second kind.
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