Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Abstract

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields R , C , H . \mathbb R, \mathbb C, \mathbb H. These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.