Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 e...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2011 |
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| Sprache: | English |
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Інститут математики НАН України
2011
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| Zitieren: | Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere / E.G. Kalnins, W. Miller Jr., S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 46 назв. — англ. |
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Kalnins, E.G. Miller Jr., W. Post, S. 2019-02-13T18:06:22Z 2019-02-13T18:06:22Z 2011 Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere / E.G. Kalnins, W. Miller Jr., S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R12; 33C45 DOI:10.3842/SIGMA.2011.051 https://nasplib.isofts.kiev.ua/handle/123456789/147168 We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials. This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html. Thanks to Jonathan Kress for valuable advice on computer verification of the dif ference operator realization for the structure formulas. S.P. acknowledges a postdoctoral IMS fellowship awarded by the Mathematical Physics Laboratory of the Centre de Recherches Math´ematiques. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere |
| spellingShingle |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere Kalnins, E.G. Miller Jr., W. Post, S. |
| title_short |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere |
| title_full |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere |
| title_fullStr |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere |
| title_full_unstemmed |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere |
| title_sort |
two-variable wilson polynomials and the generic superintegrable system on the 3-sphere |
| author |
Kalnins, E.G. Miller Jr., W. Post, S. |
| author_facet |
Kalnins, E.G. Miller Jr., W. Post, S. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147168 |
| citation_txt |
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere / E.G. Kalnins, W. Miller Jr., S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 46 назв. — англ. |
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2025-12-07T13:30:58Z |
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2025-12-07T13:30:58Z |
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