基于Abel多项式的累积量和卷积
Cumulants and convolutions via Abel polynomials
摘要：
我们提供了一个统一的多项式表达式，给出矩的累积量，反之亦然，在古典，布尔和自由设置。这是通过使用阿贝尔多项式的符号处理来完成的。作为一个副产品，我们表明，在自由累积量理论中的Pitman和Stanley的体积多项式的经典理论中的完全贝尔指数多项式的作用。利用广义Abel多项式构造了一类新的累积量，包括经典累积量、布尔累积量和自由累积量，以及由它们线性化的卷积。最后，通过本影傅里叶变换，我们陈述了布尔卷积和自由卷积之间的显式联系。
英文摘要：
We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution.
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