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

We introduce vast generalizations of the Hardy–Berndt sums.They involve higher-order Euler and/or Bernoulli functions, in which the variablesare affected by certain linear shifts. By employing the Fourier series techniquewe derive linear relations for these sums. In particular, these relations yieldreciprocity formulas for Carlitz, Rademacher, Mikolás and Apostol type generalizationsof the Hardy–Berndt sums, and give rise to generalizations for someGoldberg’s three-term relations. We also present an elementary proof for theMikolás’ linear relation and a reciprocity formula in terms of the generation function.