Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

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DAĞLI M. C., CAN M.

RAMANUJAN JOURNAL, cilt.44, sa.2, ss.301-341, 2017 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 44 Sayı: 2
  • Basım Tarihi: 2017
  • Doi Numarası: 10.1007/s11139-016-9808-y
  • Dergi Adı: RAMANUJAN JOURNAL
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.301-341
  • Anahtar Kelimeler: Eisenstein series, Dedekind sums, Bernoulli numbers and polynomials
  • Akdeniz Üniversitesi Adresli: Evet

Özet

In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.