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

TL;DR

This paper generalizes Hardy-Berndt sums using higher-order Euler and Bernoulli functions, deriving new reciprocity formulas and relations through Fourier series techniques, expanding the understanding of these classical sums.

Contribution

It introduces extensive generalizations of Hardy-Berndt sums involving shifted higher-order functions and establishes new reciprocity formulas and linear relations for various related sums.

Findings

Derived reciprocity formulas for generalized sums.

Established linear relations using Fourier series.

Provided elementary proof for Mikolás' relation.

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.