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The Ramanujan Journal

, Volume 48, Issue 2, pp 323–349 | Cite as

Transformation formulas of a character analogue of \(\log \theta _{2}(z)\)

  • Merve Çelebi BoztaşEmail author
  • Mümün Can
Article

Abstract

In this paper, transformation formulas for the function
$$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1} ^{\infty }\chi \left( n\right) \chi \left( m\right) \left( -1\right) ^{m}n^{s-1}e^{2\pi imnz/k} \end{aligned}$$
are obtained. Sums that appear in transformation formulas are generalizations of the Hardy–Berndt sums \(s_{j}(d,c), j=1,2,5\). As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented.

Keywords

Dedekind sums Hardy–Berndt sums Bernoulli and Euler polynomials 

Mathematics Subject Classification

11F20 11B68 

Notes

Acknowledgements

The authors are grateful to the referee for many valuable comments and suggestions.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAkdeniz UniversityAntalyaTurkey

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