Matrix calculus-based approach to orthogonal polynomial sequences. (English) Zbl 1443.42015
Summary: 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 [Integral Transforms Spec. Funct. 30, No. 2, 112–127 (2019; Zbl 1433.11007); Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 4, 3829–3862 (2019; Zbl 1425.11049)]. 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.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 11B83 | Special sequences and polynomials |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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