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 - <p style="text-align: justify;">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 $\alpha,\beta &gt;-1/2$ 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. &nbsp;Also, some numerical examples are included.</p>