Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Czyżycki, Tomasz; Hrivnák, Jiří; Patera, J.

doi:10.3390/sym10080354

Cited by 4 publications

(2 citation statements)

References 31 publications

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“…There are two fundamental means of calculation of the polynomials. The recurrence relations construction [13,15] is summarized for bivariate cases in this paper whereas the generating function method is developed recently in [16,17]. If the problem should be regarded as discretization of known polynomials of two continuous variables, then very few such polynomials can be discretized by the method developed in this paper.…”

Section: Introductionmentioning

confidence: 99%

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

2019

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We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified.

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Section: Introductionmentioning

confidence: 99%

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

2019

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“…• The existence and explicit forms of generating functions for the related Weyl group polynomials, developed in [41,42], further increase the relevance of the presented Chebyshev polynomial methods. The generating functions form a powerful tool for investigating symmetries and parity relations of the generated orthogonal polynomials and represent practical tool for efficient computer implementation and handling of the generated polynomials.…”

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confidence: 99%

Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Brus

Hrivnák

Motlochová

2018

Entropy

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Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

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Discrete cosine and sine transforms generalized to honeycomb lattice II. Zigzag boundaries

Hrivnák

Motlochová

2021

Journal of Mathematical Physics

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The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice with zigzag boundaries. The zigzag honeycomb point sets are constructed by subtracting the weight lattice from the refined root lattice points of the crystallographic root system A2. The two-variable (anti)symmetric orbit functions of the Weyl group of A2, discretized simultaneously on the triangular fragments of the root and weight lattices, induce a novel parametric family of zigzag extended Weyl and Hartley orbit functions. As specific linear combinations of the original orbit functions, the zigzag extended orbit functions retain the Neumann and Dirichlet boundary conditions. Three types of discrete complex Fourier–Weyl transforms and real-valued Hartley–Weyl transforms are detailed. The corresponding unitary transform matrices and interpolating behavior of the discrete transforms are exemplified.

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Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Cited by 4 publications

References 31 publications

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Discrete cosine and sine transforms generalized to honeycomb lattice II. Zigzag boundaries

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