Harmonics and spherical functions on Grassmann manifolds of rank two and two-variable analogues of Jacobi polynomials

1977 ◽  
pp. 141-154 ◽  
Author(s):  
Tom Koornwinder
Keyword(s):  
Jacobi Polynomials ◽  
Spherical Functions ◽  
Grassmann Manifolds

scholarly journals Limit transition between hypergeometric functions of type BC and type A

2013 ◽  
Vol 149 (8) ◽  
pp. 1381-1400 ◽  
Author(s):  
Margit Rösler ◽  
Tom Koornwinder ◽  
Michael Voit
Keyword(s):  
Hypergeometric Functions ◽  
Grassmann Manifold ◽  
Jacobi Polynomials ◽  
Type A ◽  
Explicit Representation ◽  
Spherical Functions ◽  
Limit Transition ◽  
Infinite Dimensional ◽  
Positive Roots ◽  
General Limit

AbstractLet ${F}_{BC} (\lambda , k; t)$ 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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