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, see e.g.[dJ], [RKV]. 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 limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp 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.

An Introduction to Dunkl Theory and Its Analytic Aspects

“…For instance, in [71] and [74], Heckman-Opdam hypergeometric functions associated with the root system A n´1 are obtained as limits of Heckman-Opdam hypergeometric functions associated with the root system BC n , when some multiplicities tend to infinity. See [73] for a similar result about generalized Bessel functions.…”

A Laplace-type representation for some generalized spherical functions of type $BC$

“…This means that X, Y = 2 C q i=1 x i y i . Figure 2 shows the multiplicities of the various roots depending on p, q and d. We set some terminology here which differs slightly from the usage in [14].…”

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

“…The Heckman-Opdam theory of hypergeometric functions associated with root systems generalizes the classical theory of spherical functions on Riemannian symmetric spaces; see [H], [HS] and [O1] for the general theory, and [NPP], [R2], [RKV], [RV1], [Sch] for some recent developments. In this paper we study these functions for the root systems of types A and BC in the noncompact case.…”

“…By the KAK-decomposition of G in the both cases above, the double coset space G//K may be identified with the Weyl chambers C A q := {t = (t 1 , · · · , t q ) ∈ R q : t 1 ≥ t 2 ≥ · · · ≥ t q } of type A and C B q := {t = (t 1 , · · · , t q ) ∈ R q : t 1 ≥ t 2 ≥ · · · ≥ t q ≥ 0} of type B respectively. In both cases, this identification occurs via a exponential mapping t → a t ∈ G from the Weyl chamber to a system of representatives a t of the double cosets in G. We now follow the notation in [RV1] and put a t = e t (1.1)…”

“…We point out that we are interested in this paper mainly in the case BC. As the A-case in the Heckman-Opdam theory appears as a limit of the BC-case for p → ∞ in some way (see [RKV], [RV1] for the details), it is not astonishing that all results in the BC-case are also available in the A-case without additional effort. In practice, all results below are proved first for the simpler A-case and then extended to the more interesting BC-case.…”