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Chebyshev polynomials, Catalan numbers, and tridiagonal matrices. (English. Russian original) Zbl 1446.39001
Theor. Math. Phys. 204, No. 1, 837-842 (2020); translation from Teor. Mat. Fiz. 203, No. 1, 3-9 (2020).
Summary: 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.
39A06 Linear difference equations
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
Full Text: DOI
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