  The main purpose of this paper is to compute all irreducible spherical
functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where
$K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is
accomplished by associating to a spherical function $\Phi$ on $G$ a matrix
valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It
is well known that there is a fruitful connection between the hypergeometric
function of Euler and Gauss and the spherical functions of trivial type
associated to a rank one symmetric pair $(G,K)$. But the relation of spherical
functions of types of dimension bigger than one with classical analysis, has
not been worked out even in the case of an example of a rank one pair. The
entries of $H$ are solutions of two systems of ordinary differential equations.
There is no ready made approach to such a pair of systems, or even to a single
system of this kind. In our case the situation is very favorable and the
solution to this pair of systems can be exhibited explicitely in terms of a
special class of generalized hypergeometric functions ${}_{p+1}F_p$.
