A Relativistic Conical Function and its Whittaker Limits
In previous work we introduced and studied a function R(a+,a−,c;v,v^) that is a generalization of the hypergeometric function ₂F₁ and the Askey-Wilson polynomials. When the coupling vector c∈C⁴ is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2v^) that generalizes the conical funct...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2011 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2011
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147993 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Relativistic Conical Function and its Whittaker Limits / S. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 43 назв. — англ. |
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Ruijsenaars, S. 2019-02-16T12:49:00Z 2019-02-16T12:49:00Z 2011 A Relativistic Conical Function and its Whittaker Limits / S. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C05; 33E30; 39A10; 81Q05; 81Q80 http://dx.doi.org/10.3842/SIGMA.2011.101 https://nasplib.isofts.kiev.ua/handle/123456789/147993 In previous work we introduced and studied a function R(a+,a−,c;v,v^) that is a generalization of the hypergeometric function ₂F₁ and the Askey-Wilson polynomials. When the coupling vector c∈C⁴ is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2v^) that generalizes the conical function specialization of ₂F₁ and the q-Gegenbauer polynomials. The function R is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser system of A₁ type, whereas the function R corresponds to BC₁, and is the joint eigenfunction of four hyperbolic Askey-Wilson type difference operators. We show that the R-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function R converges to a joint eigenfunction of the latter four difference operators. 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. We would like to thank M. Halln¨as for his interest and useful comments. We also thank several referees for pointing out additional references. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Relativistic Conical Function and its Whittaker Limits Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Relativistic Conical Function and its Whittaker Limits |
| spellingShingle |
A Relativistic Conical Function and its Whittaker Limits Ruijsenaars, S. |
| title_short |
A Relativistic Conical Function and its Whittaker Limits |
| title_full |
A Relativistic Conical Function and its Whittaker Limits |
| title_fullStr |
A Relativistic Conical Function and its Whittaker Limits |
| title_full_unstemmed |
A Relativistic Conical Function and its Whittaker Limits |
| title_sort |
relativistic conical function and its whittaker limits |
| author |
Ruijsenaars, S. |
| author_facet |
Ruijsenaars, S. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In previous work we introduced and studied a function R(a+,a−,c;v,v^) that is a generalization of the hypergeometric function ₂F₁ and the Askey-Wilson polynomials. When the coupling vector c∈C⁴ is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2v^) that generalizes the conical function specialization of ₂F₁ and the q-Gegenbauer polynomials. The function R is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser system of A₁ type, whereas the function R corresponds to BC₁, and is the joint eigenfunction of four hyperbolic Askey-Wilson type difference operators. We show that the R-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function R converges to a joint eigenfunction of the latter four difference operators.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147993 |
| citation_txt |
A Relativistic Conical Function and its Whittaker Limits / S. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 43 назв. — англ. |
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2025-12-07T13:23:08Z |
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