On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials

Gawronski, Wolfgang

doi:10.1016/0021-9045(87)90020-7

Cited by 38 publications

(28 citation statements)

References 10 publications

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“…The upper bound can apparently be obtained by a detailed study of the corresponding Riccati equation at ∞. If the 9 Consider for example the operator T = zD + D 2 + zD 3 + zD 4 . Here by Lemma 3:…”

Section: Theoremmentioning

confidence: 99%

“…Recent studies and interesting results on the asymptotic zero behaviour for these polynomials and the corresponding generalized polynomials can be found in e.g. [6,9,[11][12][13]17,18] and references therein. In [11] one can find bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class satisfying a second order differential equation, and as a corollary new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polynomials are established.…”

Section: Introductionmentioning

confidence: 99%

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On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

Bergkvist¹

2007

Journal of Approximation Theory

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In this paper we study the class of differential operators T = k j =1 Q j D j with polynomial coefficients Q j in one complex variable satisfying the condition deg Q j j with equality for at least one j. We show that if deg Q k < k then the root with the largest modulus of the nth degree eigenpolynomial p n of T tends to infinity when n → ∞, as opposed to the case when deg Q k = k, which we have treated previously in [T. Bergkvist, H. Rullgård, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153-171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of p n . Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.

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Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

Bergkvist¹

2007

Journal of Approximation Theory

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“…It is a generalization of the following well‐known limit of the zero‐counting measure for Laguerre polynomials, see, e.g. [, Theorem ]. Theorem Let

0 < a_{1, n}^{(α)} < a_{2, n}^{(α)} < \dots < a_{n, n}^{(α)} < \infty

denote the zeros of the Laguerre polynomial

L_{n}^{false(α false)}

where

α > - 1

.…”

Section: Zeros Of Exceptional Laguerre Polynomials: Proofsmentioning

confidence: 99%

Exceptional Laguerre Polynomials

Bonneux

Kuijlaars

2018

Stud Appl Math

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The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials, we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.

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“…In this way one may rederive, for example, the results on zero distributions of varying Hermite and Laguerre polynomials given in [1,4,8,9,10]. …”

Section: Examplesmentioning

confidence: 99%

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

Kuijlaars

Assche

1999

Journal of Approximation Theory

131

142

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dedicated to jaap korevaar on his 75th birthdayWe study the zeros of orthogonal polynomials p n, N , n=0, 1, ..., that are generated by recurrence coefficients a n, N and b n, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio nÂN converges, we show that the polynomials p n, N have an asymptotic zero distribution as nÂN tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distributions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes Wigert polynomials, and Lommel polynomials.

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On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials

Cited by 38 publications

References 10 publications

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

Exceptional Laguerre Polynomials

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

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