Three theorems on Vandermond matrices

We consider algebraic questions related to the discrete Fourier transform defined using symmetric Vandermonde matrices $\Lambda$. The main attention in the first two theorems is given to the development of independent formulations of the size $N\times N$ of the matrix $\Lambda$ and explicit formulas for the elements of the matrix $\Lambda$ using the roots of the equation $\Lambda^N = 1$. The third theorem considers rational functions $f(\lambda)$, $\lambda\in \mathbb{C}$, satisfying the condition of “materiality” $f(\lambda)=f(\frac{1}{\lambda})$, on the entire complex plane and related to the well-known problem of commuting symmetric Vandermonde matrices $\Lambda$ with (symmetric) three-diagonal matrices $T$. It is shown that already the first few equations of commutation and the above condition of materiality determine the form of rational functions $f(\lambda)$ and the equations found for the elements of three-diagonal matrices $T$ are independent of the order of $N$ commuting matrices. The obtained equations and the given examples allow us to hypothesize that the considered rational functions are a generalization of Chebyshev polynomials. In a sense, a similar, hypothesis was expressed recently published in “Teoreticheskaya i Matematicheskaya Fizika” by V. M. Bukhstaber et al., where applications of these generalizations are discussed in modern mathematical physics.