Reciprocity formulas for Hall-Wilson-Zagier type Hardy-Berndt sums
ACTA MATHEMATICA HUNGARICA, vol.163, no.1, pp.118-139, 2021 (SCI-Expanded)
- Publication Type: Article / Article
- Volume: 163 Issue: 1
- Publication Date: 2021
- Doi Number: 10.1007/s10474-020-01101-x
- Journal Name: ACTA MATHEMATICA HUNGARICA
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.118-139
- Keywords: Dedekind sum, Hardy–, Berndt sum, Bernoulli and Euler polynomial, CHARACTER ANALOGS, THETA-FUNCTIONS, DEDEKIND SUMS, EISENSTEIN, IDENTITIES, SERIES
- Open Archive Collection: AVESIS Open Access Collection
- Akdeniz University Affiliated: Yes
Abstract
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.