Abstract
We establish a relation between linear second-order difference equations corresponding to Chebyshev polynomials and Catalan numbers. The latter are the limit coefficients of a converging series of rational functions corresponding to the Riccati equation. As the main application, we show a relation between the polynomials \( \varphi _n(\mu)\) that are solutions of the problem of commutation of a tridiagonal matrix with the simplest Vandermonde matrix and Chebyshev polynomials.
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Notes
Images of these zeros are the roots of unity \(\lambda^n=1\) located on the unit circle.
This follows from the \(N\)-unitarity condition \(\Lambda\cdot\Lambda^*=N\cdot E\) for the matrix \(\Lambda\).
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Funding
The research of B. S. Bychkov was performed within a project supported by the Ministry of Science and Higher Education of the Russian Federation (CIS at Demidov Yaroslavl State University).
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Artisevich, A.E., Bychkov, B.S. & Shabat, A.B. Chebyshev polynomials, Catalan numbers, and tridiagonal matrices. Theor Math Phys 204, 837–842 (2020). https://doi.org/10.1134/S0040577920070016
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DOI: https://doi.org/10.1134/S0040577920070016