On Freud–Sobolev type orthogonal polynomials

Garza, Luis E.; Huertas, Edmundo J.; Marcellán, Francisco

doi:10.1007/s13370-019-00663-6

Cited by 6 publications

(15 citation statements)

References 27 publications

(38 reference statements)

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“…In [40], the authors considered a perturbation of the quartic Freud weight (w(x) = exp(−x 4 )) by the addition of a fixed charged point of mass λ at the origin. (See also [43] and the recent work in [28] for the Freudian-Sobolev case). It was shown in [40] that the semi-classical quartic Freud polynomials obey a differential equation of the form Equation ( 88), and the electrostatic model was studied.…”

Section: Properties Of Electrostatic Properties Of the Zeros For The Modified Freud-type Weightmentioning

confidence: 97%

“…For more on this, one can refer to [26,27] and the references therein. The obtained non-linear differential/difference equations for such orthogonal polynomials have considerable applications in modeling non-linear phenomena, Soliton Theory, Random matrix theory, Quantum oscillators and in the crystal structure in solid-state physics (for e.g., see the works by Chen and Its [13], Clarkson and Jordaan [11], Van Assche [10], Marcellán [28] and others).…”

Section: Introductionmentioning

confidence: 99%

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On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

Kelil

Appadu

2020

Mathematics

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Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(−cx6),c>0,α>−1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

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Section: Properties Of Electrostatic Properties Of the Zeros For The Modified Freud-type Weightmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

Kelil

Appadu

2020

Mathematics

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“…The authors in [14] considered a perturbation of quartic Freud weight (w(x) = exp(−x 4 )) by the addition of a fixed charged point of mass δ at the origin; the corresponding polynomials are Freud-type polynomials (see the recent work in [15]). For semiclassical orthogonality measure, it was shown in [14] that these polynomials obey a second-order linear differential equation of the form (1.7), and the electrostatic model is in sight as in [18].…”

Section: Application Of Eq (41) For Electrostatic Zero Distributionmentioning

confidence: 99%

On certain properties of perturbed Freud-type weight: a revisit

Kelil,

Appadu,

Arjika

2022

Preprint

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In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fullfill linear differential equation with some polynomial coefficients in their holonomic form. The aim of this work is explore certain characterizing properties of perturbed Freud type polynomials such as nonlinear recursion relations, finite moments, differential-recurrence and differential relations satisfied by the recurrence coefficients as well as the corresponding semiclassical orthogonal polynomials. We note that the obtained differential equation fulfilled by the considered semiclassical polynomials are used to study an electrostatic interpretation for the distribution of zeros based on the original ideas of Stieltjes.The following relations in [3] are valid for a semiclassical weight w with w(a) = w(b) = 0.

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“…Proof. From ( 7), (8), and applying the relationships (15)(16)(17)(18)(19), we deduce the following:…”

Section: Product Rulementioning

confidence: 99%

Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

Filipuk,

Mañas-Mañas,

Moreno-Balcázar

2020

Preprint

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We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well-known difference operators D q and ∆ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second-order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.

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On Freud–Sobolev type orthogonal polynomials

Cited by 6 publications

References 27 publications

On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

On certain properties of perturbed Freud-type weight: a revisit

Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

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