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Article

Многочлены Чебышёва, числа Каталана и трехдиагональные матрицы

Артисевич А. Е., Бычков Б. С., Шабат А. Б.

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 𝜑𝑛(𝜇) that are solutions of the problem of commutation of a tridiagonal matrix with the simplest Vandermonde matrix and Chebyshev polynomials.