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 Google™ Scholar. Others By: Arvesú, Jorge - Álvarez Nodarse, Renato - Marcellán, Francisco - Pan, K.
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 Title: Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros Author(s): Arvesú, JorgeÁlvarez Nodarse, RenatoMarcellán, FranciscoPan, K. Publisher: Elsevier Issued date: 17-Apr-1998 Citation: Journal of Computational and Applied Mathematics, 1998, vol. 90, n. 2, p. 135-156 URI: http://hdl.handle.net/10016/6405 ISSN: 0377-0427 DOI: 10.1016/S0377-0427(98)00005-3 Description: 22 pages, 4 figures.-- MSC1991 codes: 33C45; 33A65; 42C05.-- Dedicated to Professor Mario Rosario Occorsio on his 65th birthday.MR#: MR1624329 (99h:33029)Zbl#: Zbl 0924.33006 Abstract: We obtain an explicit expression for the Sobolev-type orthogonal polynomials $\{Q_n\}$ associated with the inner product $\langle p,q\rangle=\int _{-1}p(x)q(x)\rho(x)dx+A_1p(1)q(1)+B_1p(-1)q(-1)+A_2p'(1)q'(1)+B_2p'(-1)q'(-1)$, where $\rho(x)=(1-x) alpha(1+x) beta$ is the Jacobi weight function, $\alpha,\beta>-1$, $A_1,B_1,A_2,B_2\geq 0$ and $p,q\in\bold P$, the linear space of polynomials with real coefficients. The hypergeometric representation $({}_6F_5)$ and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in $[-1,1]$ is studied. Furthermore, we obtain some estimates for the largest zero of $Q_n(x)$. Such a zero is located outside the interval $[-1,1]$. We deduce its dependence on the masses. Finally, the WKB analysis for the distribution of zeros is presented. Sponsor: The research of the first author (J.A.) was supported by a grant of Ministerio de Educación y Cultura (MEC) of Spain. The research of the three first authors (J.A., R.A.N. and F.M.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB 96-0120-C03-01 and INTAS Project INTAS 93-0219 Ext. Review: PeerReviewed Publisher version: http://dx.doi.org/10.1016/S0377-0427(98)00005-3 Keywords: Orthogonal polynomialsJacobi polynomialsHypergeometric functionSobolev-type orthogonal polynomialsWKB method Rights: © Elsevier Appears in Collections: DM - GAMA - Artículos de Revistas