Reciprocity formulas for Hall–Wilson–Zagier type Hardy–Berndt sums

Abstract

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, Mikol\'{a}s 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 Mikol\'{a}s' linear relation and a reciprocity formula in terms of the generation function.