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Quasi-symmetric orthogonal polynomials on the real line: moments, quadrature rules and invariance under Christoffel modifications

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Abstract

Given \(a,b \in {\mathbb {R}},m= \min \{ a,b\}\) and \(M=\max \{a,b\},\) we consider the orthogonal polynomials associated with nontrivial positive measures \(\phi \) for which \(supp(\phi ) \subset (-\infty ,m] \cup [M,\infty )\) and \((x-a)d\phi (x)=-(x-b)d\phi (-x+a+b).\) For this class of measures, formulas in order to compute the moments, as well as formulas for the weights and nodes in the associated Gaussian quadrature rules are provided. We also show that the QD-algorithm can be applied in order to generate new orthogonal polynomials in a simple way. Several examples are given to illustrate the results obtained.

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Acknowledgements

The authors would like to thank the referee for his/her many valuable suggestions and comments which led us to improve this paper.

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Author notes

  1. Jairo S. Silva and Junior A. Pereira contributed equally to this work.

Authors and Affiliations

  1. Departamento de Matemática Aplicada, ICTE, UFTM-Universidade Federal do Triângulo Mineiro, Uberaba, MG, 38064-200, Brazil

    Daniel O. Veronese

  2. Departamento de Matemática, UFMA-Universidade Federal do Maranhão, São Luís, MA, 65080-805, Brazil

    Jairo S. Silva

  3. Departamento de Matemática e Estatística, UEPG-Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, 84030-900, Brazil

    Junior A. Pereira

Authors
  1. Daniel O. Veronese
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  2. Jairo S. Silva
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  3. Junior A. Pereira
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Correspondence to Daniel O. Veronese.

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Communicated by Baisheng Yan.

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Veronese, D.O., Silva, J.S. & Pereira, J.A. Quasi-symmetric orthogonal polynomials on the real line: moments, quadrature rules and invariance under Christoffel modifications. Comp. Appl. Math. 42, 123 (2023). https://doi.org/10.1007/s40314-023-02276-z

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  • DOI: https://doi.org/10.1007/s40314-023-02276-z

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