Volume 4, Issue 4
Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type

Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 478-488.

Published online: 2011-04

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• Abstract

In this paper we consider polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to \mathbb{C}$, defined on the space of all algebraic polynomials $\mathcal{P}$ by$$\mathcal{L}[p] =\int_{-1}^1 p(x) (1-x)^{\alpha-1/2} (1+x)^{\beta-1/2}\exp(i\zeta x)dx,$$ where $\alpha,\beta >-1/2$ are real numbers such that $\ell=|\beta-\alpha|$ is a positive integer, and $\zeta\in\mathbb{R}\backslash\{0\}$. We prove the existence of such orthogonal polynomials for some pairs of $\alpha$ and $\zeta$ and for all nonnegative integers $\ell$. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.  Also, some numerical examples are included.

• Keywords

Orthogonal polynomials, modified Jacobi weight function, recurrence relation, Gaussian quadrature rule.

• AMS Subject Headings

33C47, 41A55, 65D30

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-4-478, author = {Marija and P. Stanić and and 9562 and and Marija P. Stanić and Aleksandar and S. Cvetković and and 9563 and and Aleksandar S. Cvetković}, title = {Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2011}, volume = {4}, number = {4}, pages = {478--488}, abstract = {

In this paper we consider polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to \mathbb{C}$, defined on the space of all algebraic polynomials $\mathcal{P}$ by$$\mathcal{L}[p] =\int_{-1}^1 p(x) (1-x)^{\alpha-1/2} (1+x)^{\beta-1/2}\exp(i\zeta x)dx,$$ where $\alpha,\beta >-1/2$ are real numbers such that $\ell=|\beta-\alpha|$ is a positive integer, and $\zeta\in\mathbb{R}\backslash\{0\}$. We prove the existence of such orthogonal polynomials for some pairs of $\alpha$ and $\zeta$ and for all nonnegative integers $\ell$. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.  Also, some numerical examples are included.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m1039}, url = {http://global-sci.org/intro/article_detail/nmtma/5979.html} }
TY - JOUR T1 - Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type AU - P. Stanić , Marija AU - S. Cvetković , Aleksandar JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 478 EP - 488 PY - 2011 DA - 2011/04 SN - 4 DO - http://doi.org/10.4208/nmtma.2011.m1039 UR - https://global-sci.org/intro/article_detail/nmtma/5979.html KW - Orthogonal polynomials, modified Jacobi weight function, recurrence relation, Gaussian quadrature rule. AB -

In this paper we consider polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to \mathbb{C}$, defined on the space of all algebraic polynomials $\mathcal{P}$ by$$\mathcal{L}[p] =\int_{-1}^1 p(x) (1-x)^{\alpha-1/2} (1+x)^{\beta-1/2}\exp(i\zeta x)dx,$$ where $\alpha,\beta >-1/2$ are real numbers such that $\ell=|\beta-\alpha|$ is a positive integer, and $\zeta\in\mathbb{R}\backslash\{0\}$. We prove the existence of such orthogonal polynomials for some pairs of $\alpha$ and $\zeta$ and for all nonnegative integers $\ell$. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.  Also, some numerical examples are included.

Marija P. Stanić & Aleksandar S. Cvetković. (2020). Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type. Numerical Mathematics: Theory, Methods and Applications. 4 (4). 478-488. doi:10.4208/nmtma.2011.m1039
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