Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Czyżycki, Tomasz; Hrivnák, Jiří; Patera, Jiří

doi:10.3390/sym10080354

Open AccessArticle

Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂

by

Tomasz Czyżycki

^1,†,

Jiří Hrivnák

^2,*,†

and

Jiří Patera

^3,†

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

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

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

Submission 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.

Keywords: generating function; root system; orthogonal polynomial; Weyl group; Lie algebra

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MDPI and ACS Style

Czyżycki, T.; Hrivnák, J.; Patera, J. Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂. Symmetry 2018, 10, 354. https://doi.org/10.3390/sym10080354

AMA Style

Czyżycki T, Hrivnák J, Patera J. Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂. Symmetry. 2018; 10(8):354. https://doi.org/10.3390/sym10080354

Chicago/Turabian Style

Czyżycki, Tomasz, Jiří Hrivnák, and Jiří Patera. 2018. "Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂" Symmetry 10, no. 8: 354. https://doi.org/10.3390/sym10080354

APA Style

Czyżycki, T., Hrivnák, J., & Patera, J. (2018). Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂. Symmetry, 10(8), 354. https://doi.org/10.3390/sym10080354

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Generating Functions for Orthogonal Polynomials of A₂, C₂ and G₂

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