In this paper, we introduce vast generalizations of the Hardy-Berndt sums. They involve higher-order Euler and/or Bernoulli functions, in which the variables are affected by certain linear shifts. By employing the Fourier series technique we derive linear relations for these sums. In particular, these relations yield reciprocity formulas for Carlitz, Rademacher, Mikolas and Apostol type generalizations of the Hardy-Berndt sums, and give rise to generalizations for some Goldberg's three-term relations. We also present an elementary proof for the Mikolas' linear relation and a reciprocity formula in terms of the generation function.

中文翻译：

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Hall-Wilson-Zagier 型 Hardy-Berndt 和的互易公式

在本文中，我们介绍了 Hardy-Berndt 和的广泛推广。它们涉及高阶欧拉和/或伯努利函数，其中变量受某些线性位移的影响。通过使用傅立叶级数技术，我们可以推导出这些和的线性关系。特别是，这些关系为卡利茨、拉德马赫、米科拉斯和阿波斯托尔类型的 Hardy-Berndt 和的推广产生互易公式，并产生了对一些戈德堡三项关系的推广。我们还根据生成函数提出了 Mikolas 线性关系的基本证明和互易公式。