Zeros of Stieltjes and Van Vleck polynomials

Alam, Mahfooz

doi:10.1090/s0002-9947-1979-0534117-1

Cited by 11 publications

(8 citation statements)

References 10 publications

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“…He allowed the coefficients r j to be complex numbers with positive real parts. Alam [1] extended Marden's result. We refer to Chapter 2.9 of Marden's monograph [9] for more detailed information about Lamé's equation and to [19,17] for the connection with electrostatics of zeros of orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 69%

Lamé differential equations and electrostatics

Dimitrov

Assche

2000

Proc. Amer. Math. Soc.

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Abstract. The problem of existence and uniqueness of polynomial solutions of the Lamé differential equationwhere A(x), B(x) and C(x) are polynomials of degree p + 1, p and p − 1, is under discussion. We concentrate on the case when A(x) has only real zeros a j and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients r j in the partial fraction decomposition B(x)/A(x) = p j=0 r j /(x − a j ), we allow the presence of both positive and negative coefficients r j . The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges r j at a j is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.

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Section: Introductionmentioning

confidence: 69%

Lamé differential equations and electrostatics

Dimitrov

Assche

2000

Proc. Amer. Math. Soc.

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“…, p . Extensions of this Gauss-Lucas type theorem to cases when the residues a i are not necessarily positive real numbers as well as various other results on the location of zeros of Lamé polynomials have since been obtained [1,20,27]. In this paper we show that much more is actually true.…”

Section: Introductionmentioning

confidence: 51%

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

Borcea

2008

Journal of Approximation Theory

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We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated with the aforementioned polynomials. As a consequence we obtain new inequalities for the moments and logarithmic potentials of the corresponding root-counting measures and their weak- * limits in the semi-classical and various thermodynamic asymptotic regimes. We also prove analogous results for systems of orthogonal polynomials such as Jacobi polynomials.

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“…The situation when all the numbers β j are negative (for example, when Q 1 (z) = −Q 2 (z)) or have different signs seems to differ drastically from the latter case (see, for example [11,13]). Further interesting results on the distribution of the zeros of Van Vleck and Stieltjes polynomials under weaker assumptions on α i and β j were obtained in [1,22,23,45,46].…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Algebro-geometric aspects of Heine-Stieltjes theory

Shapiro

2010

Journal of the London Mathematical Society

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Abstract. The goal of the paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operatorsatisfies the conditions: i) r ≥ 0 and ii) deg Q k (z) = k + r we call it a non-degenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation:has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactlỳ n+r n´s uch polynomials V n,i (z) whose corresponding eigenpolynomials S n,i (z) are of degree n. We generalize a number of well-known results in this area and discuss occurring degeneracies.

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Zeros of Stieltjes and Van Vleck polynomials

Cited by 11 publications

References 10 publications

Lamé differential equations and electrostatics

Lamé differential equations and electrostatics

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

Algebro-geometric aspects of Heine-Stieltjes theory

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