This paper is dedicated to deriving novel formulae for the high-order derivatives of Chebyshev polynomials of the fifth-kind. The high-order derivatives of these polynomials are expressed in terms of their original polynomials. The derived formulae contain certain terminating ₄F₃(1) hypergeometric functions. We show that the resulting hypergeometric functions can be reduced in the case of the first derivative. As an important application—and based on the derived formulas—a spectral tau algorithm is implemented and analyzed for numerically solving the convection–diffusion equation. The convergence and error analysis of the suggested double expansion is investigated assuming that the solution of the problem is separable. Some illustrative examples are presented to check the applicability and accuracy of our proposed algorithm.

中文翻译：

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第五类切比雪夫多项式高阶导数的新公式：对流扩散方程的谱解

本文致力于推导第五类切比雪夫多项式高阶导数的新公式。这些多项式的高阶导数用它们的原始多项式表示。导出的公式包含某些终止₄ F ₃ (1) 超几何函数。我们证明，在一阶导数的情况下，所得到的超几何函数可以被简化。作为一个重要的应用，基于推导的公式，我们实现并分析了谱 tau 算法，以数值求解对流扩散方程。假设问题的解是可分离的，则研究了所建议的双重展开的收敛性和误差分析。提供了一些说明性示例来检查我们提出的算法的适用性和准确性。