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

Published online: 2011-04

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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} }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.

*Numerical Mathematics: Theory, Methods and Applications*.

*4*(4). 478-488. doi:10.4208/nmtma.2011.m1039