Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae

Abstract

We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.