Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

Abstract. We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite β-ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. In the special case of a normalized real Wishart matrix W (I n , s)/s, where n denotes the dimension and s the degrees of freedom, the rate is (log n/s) 1/4 , if n, s → ∞ with n ≤ s, and the rate is log n/n, if n, s → ∞ with n ≤ s ≤ n + K. In the latter case we also show the a.s. convergence of the nt largest eigenvalue of W (I n , s)/s to the corresponding quantile of the Marcenko-Pastur law.

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“…Finally, the remaining case t = 1 is obtained by similar arguments using Theorem 9 in Gatteschi [13].…”

Section: Theorem 33 Let λ 1 ≤ · · · ≤ λ N Denote the Eigenvalues Ofmentioning

confidence: 97%

Uniform approximation of eigenvalues in Laguerre and Hermite 𝛽-ensembles by roots of orthogonal polynomials

Dette

Imhof

2007

Trans. Amer. Math. Soc.

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“…, n, whereas the right-hand side of (45) becomes an upper bound for all, except the first few, zeros, and for all zeros if −.4999 ≤ α ≤ 1/2. The last three sections of [44] review results obtained by Luigi and others in the case where the parameter α is large compared to n, or both parameters α and n are large.…”

Section: Surveysmentioning

confidence: 97%

“…Asymptotic estimates and inequalities for the zeros λ (α) n,r of Laguerre polynomials L (α) n are reviewed in [44] and, here too, supplemented by new results. 6 There are some misprints that may distract the reader: ε 1 ( * ) at the bottom of p. 88 should be ε 1 (x * ); on p. 89, second text line, x should read x * ; and in the displayed equation that follows, the first term on the left should be multiplied by δ.…”

Section: Surveysmentioning

confidence: 99%

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

Gautschi

Giordano

2008

Numer Algor

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A good portion of Gatteschi's research publications-about 65%-is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi's and Whittaker's form. This work is reviewed here, and organized along methodological lines.

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“…Later, he contributed greatly to the use of Bessel functions as approximants for other functions; see, e.g., [9,10] and references. He often used differential equations methods.…”

Section: Introductionmentioning

confidence: 99%

Monotonic sequences related to zeros of Bessel functions

Lorch

Muldoon

2008

Numer Algor

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Abstract. In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J 0 at multiples of π, i.e., at the zeros of J 1/2 . This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.

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Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

Cited by 52 publications

References 11 publications

Uniform approximation of eigenvalues in Laguerre and Hermite 𝛽-ensembles by roots of orthogonal polynomials

Uniform approximation of eigenvalues in Laguerre and Hermite 𝛽-ensembles by roots of orthogonal polynomials

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

Monotonic sequences related to zeros of Bessel functions

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