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Symmetry 2018, 10(8), 354;

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Institute of Mathematics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic
Centre de Recherches Mathématiques et Département de Mathématiques et de Statistique, Université de Montréal, C. P. 6128 Centre Ville, Montréal, QC H3C 3J7, Canada
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 16 July 2018 / Revised: 10 August 2018 / Accepted: 14 August 2018 / Published: 20 August 2018
Full-Text   |   PDF [364 KB, uploaded 21 August 2018]


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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