Some Diophantine relations involving circular functions of rational angles

TL;DR

This paper computes eigenvalues and sums involving circular functions of rational angles, providing explicit formulas for these quantities in terms of integers, advancing understanding of Diophantine relations with trigonometric functions.

Contribution

It introduces new explicit formulas for eigenvalues and sums of circular functions at rational angles, connecting Diophantine relations with matrix eigenvalues.

Findings

Eigenvalues of specific off-diagonal matrices are explicitly calculated.

Summations involving cotangent and sine powers are expressed in simple formulas.

Results are given in terms of integers, revealing underlying number-theoretic structure.

Abstract

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