  We study the processes $K\bar{K} \to \phi$, $\pi D \to D^\ast$, $\pi \bar{D}
\to \bar{D}^\ast$, and the production of $\psi (3770)$, $\psi (4040)$, $\psi
(4160)$, and $\psi (4415)$ mesons in collisions of charmed mesons or charmed
strange mesons. The process of 2-to-1 meson-meson scattering involves a quark
and an antiquark from the two initial mesons annihilating into a gluon and
subsequently the gluon being absorbed by the spectator quark or antiquark.
Transition amplitudes for the scattering process derive from the transition
potential in conjunction with mesonic quark-antiquark wave functions and the
relative-motion wave function of the two initial mesons. We derive these
transition amplitudes in the partial wave expansion of the relative-motion wave
function of the two initial mesons so that parity and total-angular-momentum
conservation are maintained. We calculate flavor and spin matrix elements in
accordance with the transition potential and unpolarized cross sections for the
reactions using the transition amplitudes. Cross sections for the production of
$\psi (4040)$, $\psi (4160)$, and $\psi (4415)$ relate to nodes in their radial
wave functions. We suggest the production of $\psi (4040)$, $\psi (4160)$, and
$\psi (4415)$ as probes of hadronic matter that results from the quark-gluon
plasma created in ultrarelativistic heavy-ion collisions.
