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TMF, 1991, Volume 89, Number 2, Pages 190–204 (Mi tmf5886)  

This article is cited in 22 scientific papers (total in 22 papers)

“Hidden symmetry” of Askey–Wilson polynomials

A. S. Zhedanov

Abstract: A new $q$-commutator Lie algebra with three generators, $AW(3)$, is considered, and its finite-dimensional representations are investigated. The overlap functions between the two dual bases in this algebra are expressed in terms of Askey–Wilson polynomials of general form of a discrete argument: to the four parameters of the polynomials there correspond four independent structure parameters of the algebra. Special and degenerate cases of the algebra $AW(3)$ that generate all the classical polynomials of discrete arguments – Racah, Hahn, etc., – are considered. Examples of realization of the algebra $AW(3)$ in terms of the generators of the quantum algebras of $SU(2)$ and the $q$-oscillator are given. It is conjectured that the algebra $AW(3)$ is a dynamical symmetry algebra in all problems in which $q$-polynomials arise as eigenfunctions.

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English version:
Theoretical and Mathematical Physics, 1991, 89:2, 1146–1157

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Received: 14.01.1991

Citation: A. S. Zhedanov, ““Hidden symmetry” of Askey–Wilson polynomials”, TMF, 89:2 (1991), 190–204; Theoret. and Math. Phys., 89:2 (1991), 1146–1157

Citation in format AMSBIB
\by A.~S.~Zhedanov
\paper ``Hidden symmetry'' of Askey--Wilson polynomials
\jour TMF
\yr 1991
\vol 89
\issue 2
\pages 190--204
\jour Theoret. and Math. Phys.
\yr 1991
\vol 89
\issue 2
\pages 1146--1157

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    This publication is cited in the following articles:
    1. A. S. Zhedanov, “Weyl shift of $q$-oscillator and $q$-polynomials”, Theoret. and Math. Phys., 94:2 (1993), 219–224  mathnet  crossref  mathscinet  zmath  isi
    2. Sh. M. Nagiyev, “Difference Schrödinger equation and $q$-oscillator model”, Theoret. and Math. Phys., 102:2 (1995), 180–187  mathnet  crossref  mathscinet  zmath  isi
    3. Spiridonov, VP, “Poisson algebras for some generalized eigenvalue problems”, Journal of Physics A-Mathematical and General, 37:43 (2004), 10429  crossref  mathscinet  zmath  adsnasa  isi
    4. Baseilhac, P, “Deformed Dolan-Grady relations in quantum integrable models”, Nuclear Physics B, 709:3 (2005), 491  crossref  mathscinet  zmath  adsnasa  isi
    5. Boyka Aneva, “Hidden Symmetries of Stochastic Models”, SIGMA, 3 (2007), 068, 12 pp.  mathnet  crossref  mathscinet  zmath
    6. Tom H. Koornwinder, “The Relationship between Zhedanov's Algebra $AW(3)$ and the Double Affine Hecke Algebra in the Rank One Case”, SIGMA, 3 (2007), 063, 15 pp.  mathnet  crossref  mathscinet  zmath
    7. Christiane Quesne, “Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions”, SIGMA, 3 (2007), 067, 14 pp.  mathnet  crossref  mathscinet  zmath
    8. Luc Vinet, Alexei Zhedanov, “Quasi-Linear Algebras and Integrability (the Heisenberg Picture)”, SIGMA, 4 (2008), 015, 22 pp.  mathnet  crossref  mathscinet  zmath
    9. Boyka Aneva, “Tridiagonal Symmetries of Models of Nonequilibrium Physics”, SIGMA, 4 (2008), 056, 16 pp.  mathnet  crossref  mathscinet  zmath
    10. E. G. Kalnins, Willard Miller. Jr., Sarah Post, “Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D”, SIGMA, 4 (2008), 008, 21 pp.  mathnet  crossref  mathscinet  zmath
    11. Tom H. Koornwinder, “Zhedanov's Algebra $AW(3)$ and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra”, SIGMA, 4 (2008), 052, 17 pp.  mathnet  crossref  mathscinet  zmath
    12. Tatsuro Ito, Paul Terwilliger, “Double Affine Hecke Algebras of Rank 1 and the $\mathbb Z_3$-Symmetric Askey–Wilson Relations”, SIGMA, 6 (2010), 065, 9 pp.  mathnet  crossref  mathscinet
    13. Aneva B., “Exact solvability of interacting many body lattice systems”, Physics of Particles and Nuclei, 41:4 (2010), 471–507  crossref  isi
    14. Paul Terwilliger, “The Universal Askey–Wilson Algebra”, SIGMA, 7 (2011), 069, 24 pp.  mathnet  crossref  mathscinet
    15. Paul Terwilliger, “The Universal Askey–Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$”, SIGMA, 7 (2011), 099, 26 pp.  mathnet  crossref  mathscinet
    16. Odake S., Sasaki R., “Discrete quantum mechanics”, J. Phys. A: Math. Theor., 44:35 (2011), 353001  crossref  isi
    17. Vincent X. Genest, Luc Vinet, Alexei Zhedanov, “Bispectrality of the Complementary Bannai–Ito Polynomials”, SIGMA, 9 (2013), 018, 20 pp.  mathnet  crossref  mathscinet
    18. Paul Terwilliger, “The Universal Askey–Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$”, SIGMA, 9 (2013), 047, 40 pp.  mathnet  crossref  mathscinet
    19. Ernest G. Kalnins, Willard Miller Jr., Sarah Post, “Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials”, SIGMA, 9 (2013), 057, 28 pp.  mathnet  crossref  mathscinet
    20. Sarah Post, “Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$”, SIGMA, 11 (2015), 057, 17 pp.  mathnet  crossref  mathscinet
    21. Paul Terwilliger, “The $q$-Onsager Algebra and the Universal Askey–Wilson Algebra”, SIGMA, 14 (2018), 044, 18 pp.  mathnet  crossref
    22. Hadewijch De Clercq, “Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra”, SIGMA, 15 (2019), 099, 32 pp.  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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