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Symmetry, Integrability and Geometry: Methods and Applications, 2016, Volume 12, 021, 37 pp.
DOI: https://doi.org/10.3842/SIGMA.2016.021 (Mi sigma1103) 		 
This article is cited in 12 scientific papers (total in 12 papers)

Classes of Bivariate Orthogonal Polynomials

Mourad E. H. Ismailab, Ruiming Zhangc

a Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA
b Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
c College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P.R. China
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DOI: https://doi.org/10.3842/SIGMA.2016.021
Abstract: We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or $q$-difference operators.
Keywords: disc polynomials; Zernike polynomials; 2$D$-Laguerre polynomials; $q$-2$D$-Laguerre polynomials; generating functions; ladder operators; $q$-Sturm–Liouville equations; $q$-integrals; $q$-Zernike polynomials; 2$D$-Jacobi polynomials; $q$-2$D$-Jacobi polynomials; connection relations; biorthogonal functions; generating functions; Rodrigues formulas; zeros.
Funding agency Grant number
National Science Foundation 11371294
King Abdulaziz University of Saudi Arabia 14-MAT623-02
Research of M.E.H. Ismail supported by a grant from DSFP program at King Saud and by the National Plan for Science, Technology and innovation (MAARIFAH), King Abdelaziz City for Science and Technology, Kingdom of Saudi Arabia, Award number 14-MAT623-02. Research of R. Zhang partially supported by National Science Foundation of China, grant No. 11371294.
Received: August 4, 2015; in final form February 15, 2016; Published online February 24, 2016
Bibliographic databases:
Document Type: Article
MSC: 33C50; 33D50; 33C45; 33D45
Language: English
Citation: Mourad E. H. Ismail, Ruiming Zhang, “Classes of Bivariate Orthogonal Polynomials”, SIGMA, 12 (2016), 021, 37 pp.
Citation in format AMSBIB
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\paper Classes of Bivariate Orthogonal Polynomials
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    • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Symmetry, Integrability and Geometry: Methods and Applications
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