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

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CAN M.

ACTA MATHEMATICA HUNGARICA, cilt.163, sa.1, ss.118-139, 2021 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 163 Sayı: 1
  • Basım Tarihi: 2021
  • Doi Numarası: 10.1007/s10474-020-01101-x
  • Dergi Adı: ACTA MATHEMATICA HUNGARICA
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.118-139
  • Anahtar Kelimeler: Dedekind sum, Hardy&#8211, Berndt sum, Bernoulli and Euler polynomial, CHARACTER ANALOGS, THETA-FUNCTIONS, DEDEKIND SUMS, EISENSTEIN, IDENTITIES, SERIES
  • Akdeniz Üniversitesi Adresli: Evet

Özet

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.