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 Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 1, Pages 5–12 (Mi vmj710)

Three theorems on Vandermond matrices

A. E. Artisevicha, A. B. Shabatb

a Adyghe State University, 208 Pervomayskaya St., Maikop 385000, Russia
b Landau Institute for Theoretical Physics, 1A Akademika Semenova Ave., Chernogolovka 142432, Russia

Abstract: 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.

Key words: Vandermond matrix, discrete Fourier transform, commutation conditions, Laurent polynomials.

DOI: https://doi.org/10.23671/VNC.2020.1.57532

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UDC: 517.95
MSC: 42A38

Citation: A. E. Artisevich, A. B. Shabat, “Three theorems on Vandermond matrices”, Vladikavkaz. Mat. Zh., 22:1 (2020), 5–12

Citation in format AMSBIB
\Bibitem{ArtSha20} \by A.~E.~Artisevich, A.~B.~Shabat \paper Three theorems on Vandermond matrices \jour Vladikavkaz. Mat. Zh. \yr 2020 \vol 22 \issue 1 \pages 5--12 \mathnet{http://mi.mathnet.ru/vmj710} \crossref{https://doi.org/10.23671/VNC.2020.1.57532} 

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This publication is cited in the following articles:
1. A. E. Artisevich, B. S. Bychkov, A. B. Shabat, “Chebyshev polynomials, Catalan numbers, and tridiagonal matrices”, Theoret. and Math. Phys., 204:1 (2020), 837–842
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