Zero distribution of polynomials satisfying a differential-difference equation

Dominici, Diego; Assche, Walter Van

doi:10.1142/s0219530514500390

Cited by 5 publications

(7 citation statements)

References 21 publications

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“…In a previous work [16], we found the asymptotic zero distribution of polynomial families satisfying first-order differential-recurrence relations of the form (12). It would be interesting to know if our results could be extended to include the polynomials P n (c) studied in this paper.…”

Section: Discussionmentioning

confidence: 72%

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Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Dominici

2016

SIGMA

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Section: Discussionmentioning

confidence: 72%

“…In a series of papers [7,8,10,13,14,16], we studied polynomial solutions of differentialdifference equations of the form…”

Section: Introductionmentioning

confidence: 99%

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Dominici

2016

SIGMA

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“…These identities may be considered as analogues of the properties of the zeros of the Askey scheme and generalized hypergeometric polynomials proved in [12,13,14,15], for the case of the polynomial families considered in this paper. An application of the identities proved in this paper is related to the study of the asymptotic behavior of algebraic expressions involving the zeros of orthogonal polynomials of degree N as N → ∞, see [32,33].…”

Section: Generalized Pseudospectral and Spectral Matrix Representatio...mentioning

confidence: 97%

Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

Bihun

Mourning

2018

Advances in Mathematical Physics

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Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family { ] ( )} ∞ ]=0 orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations A ] ( ) = ] ( ) ] ( ), where A is a linear differential operator and each ] ( ) is a polynomial of degree at most 0 ∈ N; 0 does not depend on ]. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

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“…with A n (x) = (x 2 − 1)/b n , and B n (x) = −a n x/b n . The polynomial systems (2.3) have been studied in greater generality by Dominici et al in [13,14], where A n (x) and B n (x) are polynomials of degree at most 2 and 1 respectively. In [13] the authors study the zeros of polynomial solutions to equation (2.3), in which they analyze when their zeros are real and simple and whether the zeros of polynomials of adjacent degree interlace.…”

Section: The Differential-difference Relationmentioning

confidence: 99%

On polynomial transformations preserving purely imaginary zeros*

Hindmarsh

Lettington

2022

Integral Transforms and Special Functions

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In this present work polynomial transformations are identified that preserve the property of the polynomials having all zeros lying on the imaginary axis. Existence results concerning families of polynomials whose generalised Mellin transforms have zeros all lying on the critical line ℜs = 1 2 are then derived. Inherent structures are identified from which a simple proof relating to the Gegenbauer family of orthogonal polynomials is subsequently deduced. Some discussion about the choice of generalised Mellin transform is also given.

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Zero distribution of polynomials satisfying a differential-difference equation

Abstract: In this paper we investigate the asymptotic distribution of the zeros of polynomials P n (x) satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.MSC-class: 34E05 (Primary) 11B83, 33C45, 44A15 (Secondary)

Cited by 5 publications

References 21 publications

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

On polynomial transformations preserving purely imaginary zeros*

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