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SIGMA, 2016, Volume 12, 003, 27 pp. (Mi sigma1085)  

This article is cited in 4 scientific papers (total in 4 papers)

Doubling (Dual) Hahn Polynomials: Classification and Applications

Roy Oste

Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium

Abstract: We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester–Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

Keywords: Hahn polynomial; Racah polynomial; Christoffel pair; symmetric orthogonal polynomials; tridiagonal matrix; matrix eigenvalues; finite oscillator model.

DOI: https://doi.org/10.3842/SIGMA.2016.003

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Bibliographic databases:

ArXiv: 1507.01821
MSC: 33C45; 33C80; 81R05; 81Q65
Received: July 13, 2015; in final form January 4, 2016; Published online January 7, 2016
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Citation: Roy Oste, “Doubling (Dual) Hahn Polynomials: Classification and Applications”, SIGMA, 12 (2016), 003, 27 pp.

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. Oste, J. Van der Jeugt, “A finite oscillator model with equidistant position spectrum based on an extension of ${\mathfrak{su}}(2)$”, J. Phys. A-Math. Theor., 49:17 (2016), 175204  crossref  mathscinet  zmath  isi  elib  scopus
    2. R. Oste, J. Van der Jeugt, “Algebraic structures related to Racah doubles”, Springer Proceedings in Mathematics and Statistics, 191, 2016, 559-564  crossref  mathscinet  zmath
    3. R. Oste, J. Van der Jeugt, “Tridiagonal test matrices for eigenvalue computations: two-parameter extensions of the Clement matrix”, J. Comput. Appl. Math., 314 (2017), 30–39  crossref  mathscinet  zmath  isi  scopus
    4. R. Oste, J. Van der Jeugt, “A finite quantum oscillator model related to special sets of Racah polynomials”, Phys. Atom. Nuclei, 80:4 (2017), 786–793  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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