Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function and related functions in free probability. To be specific, we show that the function $$ \frac{\tan z}{1-x\tan z} $$ of Carlitz and Scoville describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers.

中文翻译：

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自由切线定律

Nevanlinna-Herglotz 函数在研究自由概率中的无限可分分布方面发挥着重要作用。在本文中，我们研究切线函数的作用，它是自由概率中的基本 Herglotz-Nevanlinna 函数和相关函数。具体来说，我们证明了 Carlitz 和 Scoville 的函数 $$ \frac{\tan z}{1-x\tan z} $$ 描述了自由交换子和反交换子的和的极限分布，因此给出了自由累积量由欧拉之字形数。