This paper deals with the problem of characterizing all functions $f:[0,1]\to {\mathbb{R}}_{+}$ such that $C_f(x,y)=xyf((1-x)(1-y))$, $x,y\in [0,1]$ is a bivariate copula. We provide a complete characterization for the two cases: (i) when $C_f$ is, in addition, totally positive of order 2 (TP₂) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for $C_f$ to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described.

本文处理表征所有函数的问题 $F：[0,1]\to {\mathbb{电阻}}_{+}$ 以至于 $C_F(X,是)=X是F((1-X)(1-是))$, $X,是\in [0,1]$是一个二元 copula。我们为两种情况提供了完整的表征：（i）当$C_F$此外，当f是两次连续可微时，是₂阶 (TP ₂ ) 和 (ii) 的完全正数。一般而言，函数f几乎处处都是可微的 Lebesgue 的两倍，如研究必要条件所示$C_F$成为一个copula。该论文还包含许多示例，说明获得的结果以及与文献中已知事实的联系。此外，还描述了这种 copula 的几个特性。