Identities on poly-Dedekind sums

TL;DR

This paper introduces poly-Dedekind sums, generalizing classical Dedekind sums by incorporating poly-Bernoulli functions, and establishes a reciprocity relation for these new sums, expanding the theoretical framework of modular transformations.

Contribution

It extends Dedekind sums by replacing Bernoulli functions with poly-Bernoulli functions, proving a new reciprocity relation for these generalized sums.

Findings

Established a reciprocity relation for poly-Dedekind sums.

Generalized Dedekind sums using poly-Bernoulli functions.

Expanded the theoretical understanding of modular transformation properties.

Abstract

Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.