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Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

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Abstract

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.

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Acknowledgments

We are grateful to Mehmet Cenkci and Ayhan Dil for good discussions and many helpful comments on a previous version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Akdeniz University, 07058, Antalya, Turkey

    M. Cihat Dağlı & Mümün Can

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  1. M. Cihat Dağlı
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  2. Mümün Can
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Correspondence to M. Cihat Dağlı.

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Dağlı, M.C., Can, M. Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series. Ramanujan J 44, 301–341 (2017). https://doi.org/10.1007/s11139-016-9808-y

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