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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Veröffentlicht in:Compositio mathematica Jg. 149; H. 8; S. 1381 - 1400
Hauptverfasser: Rösler, Margit, Koornwinder, Tom, Voit, Michael
Format: Journal Article
Sprache:Englisch
Veröffentlicht: London, UK London Mathematical Society 01.08.2013
Cambridge University Press
Schlagworte:
Functions (mathematics)
Geometry
Hypergeometric functions
Infinity
Manifolds
Mathematical analysis
Polynomials
Representations
Roots
53C35 (secondary)
hypergeometric functions associated with root systems
spherical functions
Grassmann manifolds
33C52
Heckman–Opdam theory
Olshanski spherical pairs
33C67 (primary)
43A90
33C80
limit transitions
ISSN:0010-437X, 1570-5846
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Zusammenfassung: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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ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X13007045