  Granular gas mixtures modeled as systems of inelastic and rough particles,
either hard disks on a plane or hard spheres, are considered. Both classes of
systems are embedded in a three-dimensional space ($d=3$) but, while in the
hard-sphere case the translational and angular velocities are vectors with the
same dimensionality (and thus there are $d_{\text{tr}}=3$ translational and
$d_{\text{rot}}=3$ rotational degrees of freedom), in the hard-disk case the
translational velocity vectors are planar (i.e., $d_{\text{tr}}=2$
translational degrees of freedom) and the angular velocity vectors are
orthogonal to the motion plane (i.e., $d_{\text{rot}}=1$ rotational degree of
freedom). This complicates a unified presentation of both classes of systems,
in contrast to what happens for smooth, spinless particles, where a treatment
of $d$-dimensional spheres is possible. In this paper, a kinetic-theory
derivation of the (collisional) energy production rates $\xi_{ij}^{\text{tr}}$
and $\xi_{ij}^{\text{rot}}$ (where the indices $i$ and $j$ label different
components) in terms of the numbers of degrees of freedom $d_{\text{tr}}$ and
$d_{\text{rot}}$ is presented. Known hard-sphere and hard-disk expressions are
recovered by particularizing to $(d_{\text{tr}},d_{\text{rot}})=(3,3)$ and
$(d_{\text{tr}},d_{\text{rot}})=(2,1)$, respectively. Moreover, in the case of
spinless particles with $d=d_{\text{tr}}$, known energy production rates
$\xi_{ij}^{\text{tr}}=\xi_{ij}$ of smooth $d$-dimensional spheres are also
recovered.
