  A conjectural formula for the $k$-point generating function of Gromov--Witten
invariants of the Riemann sphere for all genera and all degrees was proposed in
\cite{DY2}. In this paper, we give a proof of this formula together with an
explicit analytic (as opposed to formal) expression for the corresponding
matrix resolvent. We also give a formula for the $k$-point function as a sum of
$(k-1)!$ products of hypergeometric functions of one variable. We show that the
$k$-point generating function coincides with the $\epsilon\rightarrow 0$
asymptotics of the analytic $k$-point function, and also compute three more
asymptotics of the analytic function for $\epsilon\rightarrow \infty$,
$q\rightarrow 0$, $q\rightarrow\infty$, thus defining new invariants for the
Riemann sphere.
