Asymptotic Formulas for the Zeros of the Meixner Polynomials

Jin, Xuesi; Wong, R.

doi:10.1006/jath.1998.3235

Cited by 14 publications

(10 citation statements)

References 8 publications

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“…We would like to mention that our results coincide with those obtained in [8,9]. The formulas (6.9) in [8] and (2.35) in [9] are asymptotically equal to (5.1) in the present paper, while the formulas (6.27) in [8] and (4.19) in [9] are asymptotically equal to (5.2).…”

Section: Construction Of Parametrixsupporting

confidence: 89%

Global asymptotics of the Meixner polynomials

Wang¹,

Wong²

2015

The Selected Works of Roderick S C Wong

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Using the steepest descent method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou [Ann. Math. 137(1993), 295-368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals [Constr. Approx. 14(1998), 113-150].

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Section: Construction Of Parametrixsupporting

confidence: 89%

Global asymptotics of the Meixner polynomials

Wang¹,

Wong²

2015

The Selected Works of Roderick S C Wong

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“…Our three term asymptotic approximation for the large zeros is in terms of the zeros of the Airy function Ai(z). When we let λ → 0, that is, a → 1, we obtain a three term asymptotic approximation for the large zeros of the classical Meixner polynomials, and our result agrees with [6], in which a two term asymptotic approximation is given. The additional term in our approximation is surprisingly simple.…”

Section: Introductionsupporting

confidence: 83%

“…In fact, since K 2 multiplies all terms in (2.9) and the contribution of G 1 (t) to the asymptotics of S n (x) will be exponentially small, it follows from Theorem 1 in [5], that the small zeros are located approximately at x = m with an exponentially small error. In [6] the authors make the same observation for the small zeros of the classical Meixner polynomials.…”

Section: Large N and Fixed X Asymptoticsmentioning

confidence: 65%

Uniform Asymptotic Approximations for the Meixner–sobolev Polynomials

Khwaja

Daalhuis

2012

Anal. Appl.

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We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials Sn(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of Sn(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials. The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.

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“…One of the approaches is based on the steepest descend method applied to a generating function and contains many ad hoc arguments. This program was realized (yet under certain restrictions on the corresponding parameters) for Krawtchouk, Meixner and in part for Charlier polynomials, see [1,2,3,8,9]. Methods based on different ideas for bounding zeros of orthogonal polynomials can be found in [5,6,12,13].…”

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confidence: 99%