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An electrostatic model for zeros of perturbed Laguerre polynomials. (English) Zbl 1301.33012

The paper is devoted to study properties of perturbed Laguerre polynomials, which are orthogonal with respect to a perturbation of the Laguerre measure defined by \( d\nu= d \mu+ \sum_{j=1}^m a_j \delta_{c_j}\), where \(\delta_{c_j}\) is a unit mass located at the point \(c_j \in \mathbb{R} \setminus \mathbb{R}_{+}\), \(a_j\) is a positive real number, \(m\) is a positive integer and \(d\mu\) is the Laguerre measure \(d \mu(x)= x^{\alpha} e^{-x} dx,\, \alpha >-1\). First the authors obtain the expression of the Laguerre perturbed monic polynomial \(\hat{Q}_n^{\alpha}(x)\) in terms of the monic Laguerre orthogonal polynomial \(\hat{L}_n^{\alpha}(x)\) and the kernel polynomials associated with the Laguerre polynomials. Next they obtain the outer relative asymptotics, that is, the behavior of the ratio \(\frac{Q_n^{\alpha}(x)}{L_n^{\alpha}(x)}\) when \(n\) tends to \(\infty\) and \(x \in \mathbb{C} \setminus \mathbb{R}_+\), providing an independent proof from known results studied in connection with rational approximation. They also prove that \( \hat{Q}_n^{\alpha}(x)\) is a polynomial solution of a second order linear differential equation with rational functions as coefficients, which is called holonomic equation. Finally, they give an electrostatic interpretation for the distribution of the zeros of the new polynomials as the logarithmic potential interaction of the unit positive charges in the presence of an external field.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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[1] Renato Álvarez-Nodarse and Juan J. Moreno-Balcázar, Asymptotic properties of generalized Laguerre orthogonal polynomials, Indag. Math. (N.S.) 15 (2004), no. 2, 151 – 165. · Zbl 1064.41022
[2] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. · Zbl 0389.33008
[3] Herbert Dueñas and Francisco Marcellán, Laguerre-type orthogonal polynomials: electrostatic interpretation, Int. J. Pure Appl. Math. 38 (2007), no. 3, 345 – 358. · Zbl 1134.33009
[4] Herbert Dueńas, Edmundo J. Huertas, and Francisco Marcellán, Analytic properties of Laguerre-type orthogonal polynomials, Integral Transforms Spec. Funct. 22 (2011), no. 2, 107 – 122. · Zbl 1213.33017
[5] Herbert Dueñas, Edmundo J. Huertas, and Francisco Marcellán, Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials, Numer. Algorithms 60 (2012), no. 1, 51 – 73. · Zbl 1247.33014
[6] Bujar Xh. Fejzullahu and Ramadan Xh. Zejnullahu, Orthogonal polynomials with respect to the Laguerre measure perturbed by the canonical transformations, Integral Transforms Spec. Funct. 21 (2010), no. 7-8, 569 – 580. · Zbl 1215.33007
[7] F. Alberto Grünbaum, Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 189 – 194. · Zbl 0934.33012
[8] Edmundo J. Huertas, Francisco Marcellán, and Fernando R. Rafaeli, Zeros of orthogonal polynomials generated by canonical perturbations of measures, Appl. Math. Comput. 218 (2012), no. 13, 7109 – 7127. · Zbl 1246.33004
[9] Mourad E. H. Ismail, More on electrostatic models for zeros of orthogonal polynomials, Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998), 2000, pp. 191 – 204. · Zbl 0981.42015
[10] Mourad E. H. Ismail, An electrostatics model for zeros of general orthogonal polynomials, Pacific J. Math. 193 (2000), no. 2, 355 – 369. · Zbl 1011.33011
[11] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. · Zbl 1082.42016
[12] Roelof Koekoek, Generalizations of the classical Laguerre polynomials and some q-analogues, ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Dr.) – Technische Universiteit Delft (The Netherlands). · Zbl 0737.33004
[13] Tom H. Koornwinder, Orthogonal polynomials with weight function (1-\?)^{\?}(1+\?)^{\?}+\?\?(\?+1)+\?\?(\?-1), Canad. Math. Bull. 27 (1984), no. 2, 205 – 214. · Zbl 0507.33005
[14] G. L. Lopes, Convergence of Padé approximants for meromorphic functions of Stieltjes type and comparative asymptotics for orthogonal polynomials, Mat. Sb. (N.S.) 136(178) (1988), no. 2, 206 – 226, 301 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 207 – 227. · Zbl 0658.30029
[15] G. L. Lopes, Comparative asymptotics for polynomials that are orthogonal on the real axis, Mat. Sb. (N.S.) 137(179) (1988), no. 4, 500 – 525, 57 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 2, 505 – 529.
[16] F. Marcellán, A. Martínez-Finkelshtein, and P. Martínez-González, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207 (2007), no. 2, 258 – 272. · Zbl 1131.30002
[17] F. Marcellán, A. Branquinho, and J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), no. 3, 283 – 303. · Zbl 0793.33009
[18] A. Ronveaux and F. Marcellán, Differential equation for classical-type orthogonal polynomials, Canad. Math. Bull. 32 (1989), no. 4, 404 – 411. · Zbl 0685.33008
[19] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. · JFM 65.0278.03
[20] Galliano Valent and Walter Van Assche, The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: additional material, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 419 – 447. · Zbl 0856.33002
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