Some Diophantine relations involving circular functions of rational angles

Abstract

The eigenvalues of the 3 off-diagonal matrices of rank $n$ with elements $1+i cot[(j-k)π/n], sin^{-2}[(j-k)π/n]$ and $sin^{-4}[(j-k)π/n], (j=1,2,...,n, k=1,2,...,n, j\neq k)$ are computed. The sums over $k$ from 1 to $n-1$ of $cot(kπ/n) sin(2skπ/n)$ and $sin^{-p}(kπ/n) cos(2skπ/n)$ are moreover computed for $s$ integer and $p=2$ and 4. The results are given by simple formulae in terms of integers.