A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical fu...
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Autor*in:	Simon Ruijsenaars [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2011

Schlagwörter:	relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function

Übergeordnetes Werk:	In: Symmetry, Integrability and Geometry: Methods and Applications - National Academy of Science of Ukraine, 2005, 7, p 101(2011)
Übergeordnetes Werk:	volume:7, p 101 ; year:2011

Links:	Link aufrufen Link aufrufen Journal toc

Katalog-ID:	DOAJ024289914

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520			\|a In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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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allfields	(DE-627)DOAJ024289914 (DE-599)DOAJ9efc657177da412994b93910f87c797f DE-627 ger DE-627 rakwb eng QA1-939 Simon Ruijsenaars verfasserin aut A Relativistic Conical Function and its Whittaker Limits 2011 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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. relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics In Symmetry, Integrability and Geometry: Methods and Applications National Academy of Science of Ukraine, 2005 7, p 101(2011) (DE-627)501075712 (DE-600)2205586-1 18150659 nnns volume:7, p 101 year:2011 https://doaj.org/article/9efc657177da412994b93910f87c797f kostenfrei http://dx.doi.org/10.3842/SIGMA.2011.101 kostenfrei https://doaj.org/toc/1815-0659 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7, p 101 2011
spelling	(DE-627)DOAJ024289914 (DE-599)DOAJ9efc657177da412994b93910f87c797f DE-627 ger DE-627 rakwb eng QA1-939 Simon Ruijsenaars verfasserin aut A Relativistic Conical Function and its Whittaker Limits 2011 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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. relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics In Symmetry, Integrability and Geometry: Methods and Applications National Academy of Science of Ukraine, 2005 7, p 101(2011) (DE-627)501075712 (DE-600)2205586-1 18150659 nnns volume:7, p 101 year:2011 https://doaj.org/article/9efc657177da412994b93910f87c797f kostenfrei http://dx.doi.org/10.3842/SIGMA.2011.101 kostenfrei https://doaj.org/toc/1815-0659 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7, p 101 2011
allfields_unstemmed	(DE-627)DOAJ024289914 (DE-599)DOAJ9efc657177da412994b93910f87c797f DE-627 ger DE-627 rakwb eng QA1-939 Simon Ruijsenaars verfasserin aut A Relativistic Conical Function and its Whittaker Limits 2011 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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. relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics In Symmetry, Integrability and Geometry: Methods and Applications National Academy of Science of Ukraine, 2005 7, p 101(2011) (DE-627)501075712 (DE-600)2205586-1 18150659 nnns volume:7, p 101 year:2011 https://doaj.org/article/9efc657177da412994b93910f87c797f kostenfrei http://dx.doi.org/10.3842/SIGMA.2011.101 kostenfrei https://doaj.org/toc/1815-0659 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7, p 101 2011
allfieldsGer	(DE-627)DOAJ024289914 (DE-599)DOAJ9efc657177da412994b93910f87c797f DE-627 ger DE-627 rakwb eng QA1-939 Simon Ruijsenaars verfasserin aut A Relativistic Conical Function and its Whittaker Limits 2011 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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. relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics In Symmetry, Integrability and Geometry: Methods and Applications National Academy of Science of Ukraine, 2005 7, p 101(2011) (DE-627)501075712 (DE-600)2205586-1 18150659 nnns volume:7, p 101 year:2011 https://doaj.org/article/9efc657177da412994b93910f87c797f kostenfrei http://dx.doi.org/10.3842/SIGMA.2011.101 kostenfrei https://doaj.org/toc/1815-0659 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7, p 101 2011
allfieldsSound	(DE-627)DOAJ024289914 (DE-599)DOAJ9efc657177da412994b93910f87c797f DE-627 ger DE-627 rakwb eng QA1-939 Simon Ruijsenaars verfasserin aut A Relativistic Conical Function and its Whittaker Limits 2011 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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. relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics In Symmetry, Integrability and Geometry: Methods and Applications National Academy of Science of Ukraine, 2005 7, p 101(2011) (DE-627)501075712 (DE-600)2205586-1 18150659 nnns volume:7, p 101 year:2011 https://doaj.org/article/9efc657177da412994b93910f87c797f kostenfrei http://dx.doi.org/10.3842/SIGMA.2011.101 kostenfrei https://doaj.org/toc/1815-0659 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7, p 101 2011
language	English
source	In Symmetry, Integrability and Geometry: Methods and Applications 7, p 101(2011) volume:7, p 101 year:2011
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topic_facet	relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function Mathematics
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
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callnumber-first	Q - Science
author	Simon Ruijsenaars
spellingShingle	Simon Ruijsenaars misc QA1-939 misc relativistic Calogero-Moser system misc relativistic Toda system misc relativistic conical function misc relativistic Whittaker function misc Mathematics A Relativistic Conical Function and its Whittaker Limits
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topic_title	QA1-939 A Relativistic Conical Function and its Whittaker Limits relativistic Calogero-Moser system relativistic Toda system relativistic conical function relativistic Whittaker function
topic	misc QA1-939 misc relativistic Calogero-Moser system misc relativistic Toda system misc relativistic conical function misc relativistic Whittaker function misc Mathematics
topic_unstemmed	misc QA1-939 misc relativistic Calogero-Moser system misc relativistic Toda system misc relativistic conical function misc relativistic Whittaker function misc Mathematics
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title	A Relativistic Conical Function and its Whittaker Limits
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title_full	A Relativistic Conical Function and its Whittaker Limits
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title_sort	relativistic conical function and its whittaker limits
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title_auth	A Relativistic Conical Function and its Whittaker Limits
abstract	In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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.
abstractGer	In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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.
abstract_unstemmed	In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials. When the coupling vector c∈C^4 is specialized to (b,0,0,0), b∈C, we obtain a function R(a+,a−,b;v,2vˆ) that generalizes the conical function specialization of _2F_1 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 A_1 type, whereas the function R corresponds to BC_1, 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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title_short	A Relativistic Conical Function and its Whittaker Limits
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