Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation 𝔼φ (z − X − f(X)Y f∗(X))−1|X for free random variables X,Y and a Borel function f is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form X + f(X)Y f∗(X). The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

中文翻译：

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自由概率中的布尔累积量和从属关系

从属关系是自由加法和乘法卷积的分析方法的基础。我们将这种方法扩展到更一般的设置，并证明条件期望𝔼φ (z - X - F(X)是 F*(X))-1|X对于自由随机变量X,是和 Borel 函数F又是一个解决方案。该结果允许显式计算以下形式的非交换多项式的分布X + F(X)是 F*(X). 主要工具是根据布尔累积量的条件期望的新组合公式和解析器的条件期望的相应分析公式，概括了加法和乘法自由卷积的从属公式。在最后一节中，我们通过逐步的显式计算和对所有必要成分的阐述来说明结果。