  Gluons cannot directly fragment into $B_c$ mesons until at order
$\alpha_s^3$. However, the Altarelli-Parisi evolution of the $\bar b$-quark
fragmentation functions $D_{\bar b \to B_c}(z)$ and $D_{\bar b \to B_c^*}(z)$
from the heavy quark mass scales up to the collider energy scale $Q$ can induce
the gluon fragmentation functions $D_{g \to B_c}(z)$ and $D_{g \to B_c^*}(z)$,
respectively, through the gluon splitting $g\to \bar b$ at the order
$\alpha_s^3 \, {\rm log} (Q/m_{b,c})$. We will determine these induced gluon
fragmentation functions. The $B_c$ meson productions due to the induced gluon
fragmentation will be evaluated and compared to the productions by the direct
$\bar b$-quark fragmentation. The contribution from the induced gluon
fragmentation is found to be a significant portion of the total $B_c$ and
$B_c^*$ meson productions, and therefore cannot be ignored.
