ABSTRACT We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse relation about the hyperbolic sine function, which is a 2-step function of the exponential function. In this article, we show several expressions, relations, and properties about poly-Cauchy numbers with level 2. Poly-Cauchy numbers with level 2 can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form. A simple case of Cauchy numbers with level 2 has some more relations with D numbers from higher-order Bernoulli numbers or complementary Euler numbers. We prove some more expressions in determinants, continued fractions or by Trudi's formula.

中文翻译：

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具有级别 2 的多柯西数

摘要 我们引入了第 2 级的多柯西数。多柯西数可以解释为一种利用指数和对数的反比关系对经典柯西数的推广。相反，多伯努利数可以来自对数和指数的反比关系。在这个类似的流中，可以从关于双曲正弦函数的逆关系中产生具有级别 2 的多柯西数，该函数是指数函数的 2 步函数。在本文中，我们展示了关于 2 级多柯西数的几个表达式、关系和性质。 2 级多柯西数可以用组合求和的多项式系数、第一类斯特林数或迭代表示积分。我们还给出了级别为 2 的多柯西数的一些递推关系。当指数为负时，二重求和可以表述为封闭形式。一个简单的 2 级柯西数与来自高阶伯努利数或互补欧拉数的 D 数有更多关系。我们证明了一些在行列式、连分数或特鲁迪公式中的表达式。