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

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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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
\Bibitem{Zhe91} \by A.~S.~Zhedanov \paper Hidden symmetry'' of Askey--Wilson polynomials \jour TMF \yr 1991 \vol 89 \issue 2 \pages 190--204 \mathnet{http://mi.mathnet.ru/tmf5886} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1151381} \zmath{https://zbmath.org/?q=an:0782.33012|0744.33009} \transl \jour Theoret. and Math. Phys. \yr 1991 \vol 89 \issue 2 \pages 1146--1157 \crossref{https://doi.org/10.1007/BF01015906} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991HV82200003} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Sh. M. Nagiyev, “Difference Schrödinger equation and $q$-oscillator model”, Theoret. and Math. Phys., 102:2 (1995), 180–187
3. Spiridonov, VP, “Poisson algebras for some generalized eigenvalue problems”, Journal of Physics A-Mathematical and General, 37:43 (2004), 10429
4. Baseilhac, P, “Deformed Dolan-Grady relations in quantum integrable models”, Nuclear Physics B, 709:3 (2005), 491
5. Boyka Aneva, “Hidden Symmetries of Stochastic Models”, SIGMA, 3 (2007), 068, 12 pp.
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.
7. Christiane Quesne, “Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions”, SIGMA, 3 (2007), 067, 14 pp.
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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.
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.
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.
13. Aneva B., “Exact solvability of interacting many body lattice systems”, Physics of Particles and Nuclei, 41:4 (2010), 471–507
14. Paul Terwilliger, “The Universal Askey–Wilson Algebra”, SIGMA, 7 (2011), 069, 24 pp.
15. Paul Terwilliger, “The Universal Askey–Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$”, SIGMA, 7 (2011), 099, 26 pp.
16. Odake S., Sasaki R., “Discrete quantum mechanics”, J. Phys. A: Math. Theor., 44:35 (2011), 353001
17. Vincent X. Genest, Luc Vinet, Alexei Zhedanov, “Bispectrality of the Complementary Bannai–Ito Polynomials”, SIGMA, 9 (2013), 018, 20 pp.
18. Paul Terwilliger, “The Universal Askey–Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$”, SIGMA, 9 (2013), 047, 40 pp.
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.
20. Sarah Post, “Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$”, SIGMA, 11 (2015), 057, 17 pp.
21. Paul Terwilliger, “The $q$-Onsager Algebra and the Universal Askey–Wilson Algebra”, SIGMA, 14 (2018), 044, 18 pp.
22. Hadewijch De Clercq, “Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra”, SIGMA, 15 (2019), 099, 32 pp.
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