  We study the weak productions of novel heavy mesons, such as
$\eta_c^{\prime}$, $h_c$, $h_c^{\prime}$, $\chi_{c0}^{\prime}$, X(3940),
Y(3940), X(3872), and Y(4260), in the semi-leptonic $B_c$ decays. Since there
is still no definite answer for the components of X(3940), Y(3940), X(3872),
Y(4260) so far, we will assign them as excited charmonium states with the
possible quantum numbers constrained by the current experiments. As for the
weak transition form factors, we calculate them in the framework of light-cone
QCD sum rules approach, which is proved to be a powerful tool to deal with the
non-perturbative hadronic matrix element. Our results indicate that different
interpretations of X(3940) can result in remarkable discrepancy of the
production rate in the $B_c$ decays, which would help to clarify the inner
structure of the X(3940) with the forthcoming LHC-b experiments. Besides, the
predicted large weak production rates of X(3872) and Y(3940) in $B_c$ decays
and the small semi-leptonic decay rate for $B_c \to Y(4260)$ all depend on
their quantum number $J^{PC}$ assignments. Moreover, the $S-D$ mixing of
various vector charmonium states in the weak decay of $B_c$ is also discussed
in this work. The future experimental measurements of these decays will test
the inner structures of these particles, according to our predictions here.
