论文标题
Hall-Wilson-Zagier类型Hardy-Berndt总和的互惠公式
Reciprocity formulas for Hall-Wilson-Zagier type Hardy-Berndt sums
论文作者
论文摘要
在本文中，我们介绍了Hardy-Berndt总和的广泛概括。它们涉及高阶Euler和/或Bernoulli函数，其中变量受某些线性移位的影响。通过采用傅立叶系列技术，我们得出这些总和的线性关系。特别是，这些关系产生了carlitz，Rademacher，Mikolás和Apostol类型的概括的互惠公式，并对Hardy-Berndt总和进行了概括，并引起了一些Goldberg的三个期关系的概括。我们还为Mikolás的线性关系和互惠公式提供了基本证明。
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, Mikolás 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 Mikolás' linear relation and a reciprocity formula in terms of the generation function.
