  We study the action of hypergeometric shift operators on the Heckman-Opdam
hypergeometric functions associated with the $BC_n$ type root system and some
negative multiplicities. Those hypergeometric functions are connected to the
$\chi$-spherical functions on Hermitian symmetric spaces $U/K$ where $\chi$ is
a nontrivial character of $K$. We apply shift operators to the hypergeometric
functions to move negative multiplicities to positive ones. This allows us to
use many well-known results of the hypergeometric functions associated with
positive multiplicities. In particular, we use this technique to achieve
exponential estimates for the $\chi$-spherical functions. The motive comes from
the Paley-Wiener type theorem on line bundles over Hermitian symmetric spaces.
