Geodesic
Three theorems on Vandermond matrices
Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 5-12 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. E. Artisevich; A. B. Shabat. Three theorems on Vandermond matrices. Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 5-12. http://geodesic.mathdoc.fr/item/VMJ_2020_22_1_a0/

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