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Symmetry 2018, 10(8), 354; https://doi.org/10.3390/sym10080354

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

1
Institute of Mathematics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland
2
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic
3
Centre de Recherches Mathématiques et Département de Mathématiques et de Statistique, Université de Montréal, C. P. 6128 Centre Ville, Montréal, QC H3C 3J7, Canada
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 16 July 2018 / Revised: 10 August 2018 / Accepted: 14 August 2018 / Published: 20 August 2018
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Abstract

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials. View Full-Text
Keywords: generating function; root system; orthogonal polynomial; Weyl group; Lie algebra generating function; root system; orthogonal polynomial; Weyl group; Lie algebra
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Czyżycki, T.; Hrivnák, J.; Patera, J. Generating Functions for Orthogonal Polynomials of A2, C2 and G2. Symmetry 2018, 10, 354.

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