In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials.

中文翻译：

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基于矩阵微积分的正交多项式序列方法

本文提出了一种基于矩阵演算的正交多项式方法。给出了已知和新的基本结果，例如递归关系和行列式。提出了与切比雪夫算法相似但不完全相同的新算法，用于多项式的实际计算。已经考虑了单项和对称正交多项式序列的情况以及正交多项式序列的情况。给出了一些经典和非经典的例子。这些工作是从作者已经开始的更广泛的角度出发进行的。它提供了多项式的一般序列的性质的确定，因此可以确定它们对最重要的多项式的特殊类别的适用性。