On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials |
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Authors: | Jorge Arvesú Francisco Marcellán Renato Álvarez-Nodarse |
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Institution: | (1) Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés, Madrid, Spain;(2) Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080 Seville, Spain |
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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 U
U=J
, +A
1 (x–1)+B
1 (x+1)–A
2![delta](/content/7bwl24ul8tdg1wjj/xxlarge948.gif) (x–1)–B
2![delta](/content/7bwl24ul8tdg1wjj/xxlarge948.gif) (x+1), where J
, is the Jacobi linear functional, i.e. J
, ,p›= –1
1
p(x)(1–x) (1+x)![beta](/content/7bwl24ul8tdg1wjj/xxlarge946.gif) dx,![agr](/content/7bwl24ul8tdg1wjj/xxlarge945.gif) , >–1, p P, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C –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 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 ±1. The denominators of the n–1/n] Padé approximants are our orthogonal polynomials. |
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Keywords: | semiclassical orthogonal polynomials asymptotics zeros |
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