In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function $$\frac{tanz}{1-xtanz}$$ describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville (see (1.6) in [L. Carlitz and R. Scoville, Tangent numbers and operators, Duke Math. J. 39 (1972) 413–429]) which arose in connection with the enumeration of certain permutations. In this paper, we continue to study the limit of weighted sums of Boolean commutators and anticommutators and we show that the shifted generalized tangent function appears in a limit theorem. In order to do this, we shall provide an arbitrary cumulants formula of the quadratic form. We also apply this result to obtain several results in a Boolean probability theory.

在[W. Ejsmont 和 F. Lehner，自由切线定律，Adv。应用。数学。 121 (2020) 102093]，我们研究了自由换向器和反换向器的极限和，并表明广义正切函数$$\frac{晒黑z}{1-X晒黑z}$$描述极限分布。这是 Carlitz 和 Scoville 的高阶正切数的生成函数（参见 [L. Carlitz and R. Scoville, Tangent Numbers and Operators, Duke Math. J.  39 (1972) 413–429] 中的 (1.6)），其中与某些排列的枚举有关。在本文中，我们继续研究布尔可交换子和反可交换子的加权和的极限，并证明移动广义正切函数出现在极限定理中。为了做到这一点，我们将提供一个二次形式的任意累积量公式。我们还应用这个结果来获得布尔概率论中的几个结果。