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Three theorems on Vandermond matrices

A. E. Artisevich^a, A. B. Shabat<sup>b

a</sup> 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.

UDC: 517.95

MSC: 42A38

Received: 16.07.2019

DOI: 10.23671/VNC.2020.1.57532