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It is well known that the spherical functions of symmetric
spaces of rank one are either Bessel functions, Jacobi functions, or
Jacobi polynomials. Moreover, the product formulas and the Harish-Chandra
integral representations for the spherical functions lead by analytic
continuation to corresponding formulas for a continuous range of
these special functions.
In the talk we review recent extensions of these classical results to the
multivariate setting, where the spherical functions of Cartan motion groups
and (compact and non-compact) Grassmann manifolds are related to
Bessel
functions on Welyl chambers and to
Heckman-Opdam hypergeometric functions (and polynomials)
of type BC. We also consider some reductive cases as well as Laguerre functions
associated with Heisenberg groups.
We emphazize that explicit versions
of the integral representations lead to interesting limit transitions for
these special functions with explicit error estimates. These limits have
applications e.g. to central limit theorems for random walks on the associated
symmetric spaces and also to the spherical functions of associated Olshanski
spherical pairs of finite rank.
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