Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Horozov, Emil

doi:10.3842/sigma.2016.050

Cited by 6 publications

(16 citation statements)

References 49 publications

(90 reference statements)

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“…It is easy to check that [∆,xD −1 ] = 1. In this way we obtain VOP, which extend the results of [39,11,28] on Charlier and Meixner type polynomials.…”

Section: Introductionsupporting

confidence: 83%

“…The continuous polynomial systems can be obtained by using the standard representation of the Weyl algebra as the algebra of differential operators with polynomial coefficients in one variable x. Namely we put Y to be the operator of multiplication by x and Z to be the differentiation ∂ x . In this way we obtain extensions of the Hermite and the Laguerre polynomials, similar to those obtained in [38,22,10,11].…”

Section: Introductionsupporting

confidence: 56%

“…The present paper generalizes the results of [38,39]. There we sketched a program aiming to construct sets of Vector Orthogonal Polynomials (VOP), with the property that the polynomials of each system are eigenfunctions of some differential or difference operator.…”

Section: Introductionmentioning

confidence: 87%

“…It is natural to call the polynomials defined via q(∆) Charlier-Appell polynomials, cf. [39] and Example 7.7. In the case when q(G) = a∂ l x , a ∈ C, the polynomials are the well known Gould-Hopper polynomials [29].…”

Section: Discrete Vs Continous Vopmentioning

confidence: 99%

“…Here the new families are almost all. Only the extension of the Charlier ( [12,66] and Meixner polynomials [39] are known.…”

Section: Introductionmentioning

confidence: 99%

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Vector Orthogonal Polynomials with Bochner’s Property

Horozov

2017

Constr Approx

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Abstract. Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a classical theorem by Bochner they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard-Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem and this characterizes them.Motivated by Bochner's theorem we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner's property.The goal of this paper is to introduce methods for construction of VOP which have Bochner's property. The methods are purely algebraic and are based on automorphisms of non-commutative algebras. They also use ideas from the so called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations.To Boris Shapiro on occasion of his 60-th birthday 2010 Mathematics Subject Classification. 34L20 (Primary); 30C15, 33E05 (Secondary).

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“…It is easy to check that [∆,xD −1 ] = 1. In this way we obtain VOP, which extend the results of [39,11,28] on Charlier and Meixner type polynomials.…”

Section: Introductionsupporting

confidence: 83%

Section: Introductionsupporting

confidence: 56%

Section: Introductionmentioning

confidence: 87%

Section: Discrete Vs Continous Vopmentioning

confidence: 99%

“…Here the new families are almost all. Only the extension of the Charlier ( [12,66] and Meixner polynomials [39] are known.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Vector Orthogonal Polynomials with Bochner’s Property

Horozov

2017

Constr Approx

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Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Feigin

Hallnäs

Веселов

2021

Commun. Math. Phys.

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Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

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On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

Zagorodnyuk

2021

Journal of Difference Equations and Applications

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Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Cited by 6 publications

References 49 publications

Vector Orthogonal Polynomials with Bochner’s Property

Vector Orthogonal Polynomials with Bochner’s Property

Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

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