Some new reciprocity formulas for generalized Dedekind sums
The author considers generalized Dedekind sums of the form \[ C(r,s,h,k,u,v)=\sum_{\mu \quad mod k}P_ r((u+\mu)/k)\quad P_ s(v+(u+\mu)h/k) \] where \(P_ r(x)\) is the periodic Bernoulli function given by the usual Fourier series. Of the six parameters r,s,h,k,u,v, the first two are nonnegative integers, h is an arbitrary integer, k is a positive integer, and u,v are arbitrary real numbers. The purpose of the paper is to deduce a general reciprocity law for these sums which includes previously discovered reciprocity laws as special cases. The lengthy proof, requiring consideration of seven cases, is based primarily on rearrangements of Fourier series. A further generalization is indicated in which the Bernoulli functions are multiplied by certain roots of unity.
scientific article; zbMATH DE number 3922747
scientific article