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 Google™ Scholar. Others By: Arvesú, Jorge - Marcellán, Francisco - Álvarez Nodarse, Renato
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 Title: On a modification of the Jacobi linear functional: Asymptotic properties and zeros of orthogonal polynomials Author(s): Arvesú, JorgeMarcellán, FranciscoÁlvarez Nodarse, Renato Publisher: Springer Issued date: Apr-2002 Citation: Acta Applicandae Mathematicae, 2002, vol. 71, n. 2, p. 127-158 URI: http://hdl.handle.net/10016/6007 ISSN: 0167-8019 (Print)1572-9036 (Online) DOI: 10.1023/A:1014510004699 Description: 32 pages, 2 figures.-- MSC2000 codes: 33C45, 42C05.MR#: MR1914739 (2003e:33014)Zbl#: Zbl 1001.33008 Abstract: The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional ${\scr U}$ $${\scr U}={\scr J}_{\alpha,\beta}+A_1\delta(x-1)+B_1\delta(x+1)- A_2\delta'(x-1)-B_2\delta'(x+1),$$ where ${\scr J}_{\alpha,\beta}$ is the Jacobi linear functional, i.e. $$\langle{\scr J}_{\alpha,\beta},p\rangle=\int _{-1}p(x)(1-x) alpha (1+x) beta dx,\quad \alpha,\beta>-1,\ p\in {\Bbb P},$$ and ${\Bbb P}$ is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in $(-1,1)$ (inner asymptotics) and ${\Bbb C}\sbs [-1,1]$ (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional ${\scr U}$ is a generalization of one studied by T. H. Koornwinder when $A_2=B_2=0$. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi-Markov function by a rational function with two double poles at $\pm 1$. The denominators of the $[n-1/n]$ Padé approximants are our orthogonal polynomials. Sponsor: The first author (J.A.) was partially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM2000-0029 and BFM2000-0206-C04-01. The research of the authors (F.M. and R.A.N.) was partially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM2000-0206-C04-01 and BFM2000-0206-C04-02, respectively. In addition, the first author (J.A.) thanks Dirección General de Investigación de la Comunidad Autónoma de Madrid for its financial support. Review: PeerReviewed Publisher version: http://dx.doi.org/10.1023/A:1014510004699 Keywords: Semiclassical orthogonal polynomialsAsymptoticsZeros Rights: © Springer Appears in Collections: DM - GAMA - Artículos de Revistas