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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| Дата: | 2011 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2011
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.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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irk-123456789-1479932019-02-17T01:23:24Z A Relativistic Conical Function and its Whittaker Limits Ruijsenaars, S. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147993 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Ruijsenaars, S. |
| spellingShingle |
Ruijsenaars, S. A Relativistic Conical Function and its Whittaker Limits Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Ruijsenaars, S. |
| author_sort |
Ruijsenaars, S. |
| title |
A Relativistic Conical Function and its Whittaker Limits |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:28:21Z |
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