# Transformation laws for generalized Dedekind sums associated to Fuchsian groups

## TL;DR

This paper derives transformation laws for generalized Dedekind sums linked to Eisenstein series and their variants, applicable to various Fuchsian groups, expanding understanding of their symmetry properties and specific group cases.

## Contribution

It introduces new transformation laws for generalized Dedekind sums associated with Fuchsian groups, including non-holomorphic Eisenstein series and higher-order variants.

## Key findings

- Transformation laws established for generalized Dedekind sums
- Applicable to diverse Fuchsian groups including Hecke and non-congruence groups
- Examples provided for specific groups like $ ext{Hecke}$ and $ ext{Gamma}_0(N)$

## Abstract

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.