Examples of Orthogonal Polynomials in Several Variables

Dunkl, Charles F.; Xu, Yuan

doi:10.1017/cbo9781107786134.007

Cited by 66 publications

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“…, M d,l , is an orthonormal basis of L 2 (µ). For further properties of harmonic polynomials, see [1,7]. Recall that the Jacobi polynomials are defined as…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

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Fractional Laplace Operator and Meijer G-function

2016

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We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x| 2 , or generalized hypergeometric functions of −|x| 2 , multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1 − |x| 2 ) α/2 + (−∆) α/2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper [9].

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“…, M d,l , is an orthonormal basis of L 2 (µ). For further properties of harmonic polynomials, see [1,7]. Recall that the Jacobi polynomials are defined as…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

“…This system of polynomials forms a complete orthogonal system in L 2 (w), the weighted L 2 space with weight function w(x) = (1 − |x| 2 ) α/2 + , see Proposition 2.3.1 in [7]. Finally, we define…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

Fractional Laplace Operator and Meijer G-function

2016

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“…The most convenient method is the recurrence relation in Equation (2). The method of construction in [5] is complicated and time-consuming, since many computations are required.…”

Section: Legendre-weighted Orthogonal Polynomialsmentioning

confidence: 99%

“…They are well-studied, and a lot of research has been done for the univariate case, see [1]. Orthogonal polynomials over a square region can be constructed using the tensor product of univariate orthogonal polynomials, see [2,3]. Orthogonal polynomials over triangular domains have to be determined in a different way, see [4].…”

Section: Introductionmentioning

confidence: 99%

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Rababah

2016

Mathematics

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Abstract:In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n,r (u, v, w), r = 0, 1, . . . , n, n ≥ 0 on the triangular domain T = {(u, v, w) : u, v, w ≥ 0, u + v + w = 1} are constructed, where u, v, w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.

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“…In the present paper, we provide a new application of the Fourier-Wigner transform in the context of the complex Hermite polynomials H m,n [7,10,4]. More precisely, we realize H m,n (z;z) as the Fourier-Wigner transform of the well-known real Hermite functions h n on R. This reduces considerably the Wong's proof [19,Chapter 21] giving the explicit expression of V(h n , h m ) in terms of the Laguerre polynomials.…”

Section: Introductionmentioning

confidence: 99%

Complex Hermite functions as Fourier–Wigner transform

Agorram

Benkhadra

Hamyani

et al. 2015

Integral Transforms and Special Functions

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ABSTRACT. We prove that the complex Hermite polynomials H m,n on the complex plane C can be realized as the Fourier-Wigner transform V of the well-known real Hermite functions h n on real line R. This reduces considerably the Wong's proof [19, Chapter 21] giving the explicit expression of V(h m , h n ) in terms of the Laguerre polynomials. Moreover, we derive a new generating function for the H m,n as well as some new integral identities.

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Examples of Orthogonal Polynomials in Several Variables

Cited by 66 publications

References 0 publications

Fractional Laplace Operator and Meijer G-function

Fractional Laplace Operator and Meijer G-function

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Complex Hermite functions as Fourier–Wigner transform

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