Explicit evaluations and reciprocity theorems for finite trigonometric sums
For given integers \(m\geq 0\), \(h>0\), \(k>0\), and given real parameters \(a,b\) in the interval \([0,1)\), the authors consider trigonometric sums of the form \[ s_m(h,k;a,b)= \sum\cot^m \biggl(\pi \frac{r+b}{k} \biggr) \cot\Biggl(\pi \biggl( \frac{h(r+b)}{k}-a \biggr)\Biggr), \] where the sum is extended over \(0<r+b<k\). Using residue calculus they obtain a reciprocity law expressing \(s_m(h,k;a,b)/k+ s_m(k,h;b,a)/h\) in closed form in terms of polynomials evaluated at \(\cot(\pi a)\) and \(\cot(\pi b)\). Special cases include classical reciprocity laws for Dedekind sums and some of their generalizations, as well as some explicit evaluations of trigonometric sums. The paper also gives a valuable history and an extensive bibliography of evaluations and reciprocity theorems for trigonometric sums.
scientific article; zbMATH DE number 1892331
scientific article