Limit transition between hypergeometric functions of type BC and type A
Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $...
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| Vydané v: | Compositio mathematica Ročník 149; číslo 8; s. 1381 - 1400 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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London, UK
London Mathematical Society
01.08.2013
Cambridge University Press |
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| ISSN: | 0010-437X, 1570-5846 |
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| Abstract | Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski. |
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| AbstractList | Abstract Let [formula omitted, refer to PDF] be the Heckman-Opdam hypergeometric function of type BC with multiplicities [formula omitted, refer to PDF] and weighted half-sum [formula omitted, refer to PDF] of positive roots. We prove that [formula omitted, refer to PDF] converges as [formula omitted, refer to PDF] and [formula omitted, refer to PDF] to a function of type A for [formula omitted, refer to PDF] and [formula omitted, refer to PDF]. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields [formula omitted, refer to PDF] when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski. [PUBLICATION ABSTRACT] Let be the Heckman-Opdam hypergeometric function of type BC with multiplicities and weighted half-sum of positive roots. We prove that converges as and to a function of type A for and . This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski. Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski. Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $ . This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski. |
| Author | Rösler, Margit Koornwinder, Tom Voit, Michael |
| Author_xml | – sequence: 1 givenname: Margit surname: Rösler fullname: Rösler, Margit email: roesler@math.upb.de organization: Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany email roesler@math.upb.de – sequence: 2 givenname: Tom surname: Koornwinder fullname: Koornwinder, Tom email: T.H.Koornwinder@uva.nl organization: Korteweg de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 CE Amsterdam, The Netherlands email T.H.Koornwinder@uva.nl – sequence: 3 givenname: Michael surname: Voit fullname: Voit, Michael email: michael.voit@math.tu-dortmund.de organization: Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany email michael.voit@math.tu-dortmund.de |
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| Cites_doi | 10.4153/CJM-1997-019-9 10.1016/0001-8708(89)90015-7 10.1090/conm/190/2310 10.1016/j.jfa.2009.12.007 10.1007/s002200050161 10.1007/s00039-008-0658-7 10.1007/BF01076358 10.1007/BF01239517 10.1007/BF02392487 |
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| References | Rösler (S0010437X13007045_r22) 2013 Dawson (S0010437X13007045_r3) 2013; 23 Heckman (S0010437X13007045_r10) 1994 Gunning (S0010437X13007045_r5) 1990 Olshanskii (S0010437X13007045_r17) 1990 van Diejen (S0010437X13007045_r27) 1995; 95 S0010437X13007045_r8 S0010437X13007045_r1 Titchmarsh (S0010437X13007045_r26) 1939 Opdam (S0010437X13007045_r19) 2000 S0010437X13007045_r24 S0010437X13007045_r23 S0010437X13007045_r25 Heckman (S0010437X13007045_r7) 1987; 64 Faraut (S0010437X13007045_r4) 2008 Hallnäs (S0010437X13007045_r6) 2009; 9 S0010437X13007045_r20 Lassalle (S0010437X13007045_r13) 1991; 312 Macdonald (S0010437X13007045_r15) 2000; 45 Heckman (S0010437X13007045_r9) 1997; 245 S0010437X13007045_r18 Rösler (S0010437X13007045_r21) 2008; 4 S0010437X13007045_r12 Beerends (S0010437X13007045_r2) 1993; 339 Koornwinder (S0010437X13007045_r11) 1984 S0010437X13007045_r14 S0010437X13007045_r16 |
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| Snippet | Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted... Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted... Abstract Let [formula omitted, refer to PDF] be the Heckman-Opdam hypergeometric function of type BC with multiplicities [formula omitted, refer to PDF] and... Let be the Heckman-Opdam hypergeometric function of type BC with multiplicities and weighted half-sum of positive roots. We prove that converges as and to a... |
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| SubjectTerms | Functions (mathematics) Geometry Hypergeometric functions Infinity Manifolds Mathematical analysis Polynomials Representations Roots |
| Title | Limit transition between hypergeometric functions of type BC and type A |
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