AI 中文总结
本文引入由有理生成函数定义的新型广义切比雪夫多项式，证明经典四类切比雪夫多项式是其特例，推导了该多项式的递推关系、行列式表示等，建立了与其他经典序列的联系并给出相关连分数展开式。
AI 中文摘要
本文引入并研究一类由有理生成函数定义的新型广义切比雪夫多项式族。研究表明，经典的第一、第二、第三、第四类切比雪夫多项式可自然作为该框架的特例出现。此外，本文推导了这类多项式的完整递推关系、三对角行列式表示及显式闭式表达式，还通过第二类斯特林数建立了该广义多项式与其他经典序列（包括摩根-沃伊塞多项式、斐波那契多项式、富比尼多项式）的联系。最后，本文考察了相关的欧拉-塞德尔矩阵，并给出了相邻多项式项商的连分数展开式。
英文摘要
In this paper, we introduce and study a novel family of generalized Chebyshev polynomials defined via a rational generating function. We demonstrate that the classical Chebyshev polynomials of the first, second, third, and fourth kinds naturally emerge as special cases of this framework. Furthermore, we derive comprehensive recurrence relations, tridiagonal determinant representations, and explicit closed-form expressions for these polynomials. We also establish some connections between the generalized Chebyshev polynomials and other classical sequences, including Morgan-Voyce polynomials, Fibonacci polynomials, and Fubini polynomials via Stirling numbers of the second kind. Finally, we examine the associated Euler-Seidel matrix and provide a continued fraction expansion for the quotient of consecutive polynomial terms.
Comments10 pages
