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.

中文翻译：

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Chebyshev 多项式、加泰罗尼亚数和三对角矩阵

我们建立对应于 Chebyshev 多项式的线性二阶差分方程和 Catalan 数之间的关系。后者是对应于 Riccati 方程的收敛系列有理函数的极限系数。作为主要应用，我们展示了多项式 $$ \varphi _n(\mu)$$ 之间的关系，这些多项式是三对角矩阵与最简单的范德蒙矩阵和切比雪夫多项式的对易问题的解决方案。