Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

TL;DR

This paper derives transformation formulas for generalized Eisenstein series involving periodic Dedekind sums, proves reciprocity theorems, and explores their applications to infinite series evaluations and special cases.

Contribution

It introduces generalized Dedekind sums with periodic Bernoulli functions and establishes their reciprocity and transformation properties under modular substitutions.

Findings

Derived transformation formulas for generalized Eisenstein series.

Proved reciprocity theorems for the new Dedekind sums.

Connected these sums to evaluations of infinite series.

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.