Limit transition between hypergeometric functions of type BC and type A

摘要

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 tin R ^n and lambda in 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 F = R, C, 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.