Some generalized hypergeometric d-orthogonal polynomial sets

Cheikh, Youssèf Ben; Ouni, A.

doi:10.1016/j.jmaa.2008.01.055

Cited by 22 publications

(10 citation statements)

References 32 publications

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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%

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%

Vector Orthogonal Polynomials with Bochner’s Property

Horozov

2017

Constr Approx

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“…This gave us two characterization theorems dealing with q-polynomials analogous to the q-Meixner, big q-Laguerre, little q-Laguerre and q-Laguerre ones. The obtained d -orthogonal polynomials can be viewed as a q-extension of the d -orthogonal polynomials of Meixner type [14] and Laguerre type [10], since we rediscovered some properties of these polynomials for the limiting case q D 1. We summarize this point of view in the scheme given in Figure 1.…”

Section: Concluding Remarkmentioning

confidence: 99%

“…They will be defined below and have seen various applications [5,19,20,35,37,38]. The dorthogonality has been intensively investigated to prove classification and characterizations of hypergeometric and basic hypergeometric d -orthogonal polynomials similarly to the case of classical orthogonality (see, for instance, [6][7][8][9][10][11][12][13][14][15][16][17][21][22][23][24][25][28][29][30]32,41]). As an example of d -orthogonal polynomials we cite the Laguerre type [7,10] and Meixner type [12,14] ones given respectively bỳ (see [31]) with .a/ j D .aCj / .a/ .…”

Section: Introductionmentioning

confidence: 99%

d-Orthogonality of some basic hypergeometric polynomials

Lamiri

Ouni

2013

Georgian Mathematical Journal

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In this paper, we give two characterization theorems to introduce new examples of basic hypergeometric d -orthogonal polynomials which, on the one hand, generalize the known q-Meixner, big q-Laguerre, little q-Laguerre and q-Laguerre polynomials and, on the other hand, can be viewed as q-analogs of d -orthogonal polynomials of Meixner and Laguerre type. For the resulting polynomials, we state some properties.

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“…The classification of d-orthogonal polynomials that have a hypergeometric representation of the form (1.2) has been studied recently in [3]. The results are as follows.…”

Section: D-orthogonal Polynomials As Generalized Hypergeometric Funct...mentioning

confidence: 99%

d-Orthogonal polynomials and su(2)

Genest,

Vinet,

Zhedanov

2011

Preprint

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Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows for their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.). In the limit where su(2) contracts to the Heisenberg-Weyl algebra h 1 , the polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.

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Some generalized hypergeometric d-orthogonal polynomial sets

Cited by 22 publications

References 32 publications

Vector Orthogonal Polynomials with Bochner’s Property

Vector Orthogonal Polynomials with Bochner’s Property

d-Orthogonality of some basic hypergeometric polynomials

d-Orthogonal polynomials and su(2)

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