  For a connected semisimple real Lie group $G$ of non-compact type, Wallach
introduced a class of $K$-types called small. We classify all small $K$-types
for all simple Lie groups and prove except just one case that each elementary
spherical function for each small $K$-type $(\pi,V)$ can be expressed as a
product of hyperbolic cosines and a Heckman-Opdam hypergeometric function. As
an application, the inversion formula for the spherical transform on $G\times_K
V$ is obtained from Opdam's theory on hypergeometric Fourier transforms.
