  The recent measurements on $R_K$ and $R_{\pi}$ imply that there exists a
possible violation of the leptonic flavor universality which is one of the
cornerstones of the standard model. It is suggested that a mixing between
sterile and active neutrinos might induce such a violation. In this work we
consider the scenarios with one or two sterile neutrinos to explicitly realize
the data while the constraints from the available experiments have been taken
into account. Moreover, as indicated in literature, the deviation of the real
PMNS matrix from the symmetric patterns may be due to a $\mu-\tau$ asymmetry,
therefore the measurements on $R_{D(D_s)e\mu} = \Gamma(D(D_s)\rightarrow
e^+\nu_e)/\Gamma(D(D_s)\rightarrow \mu^+\nu_\mu)$ and $R_{D(D_s)\mu\tau} =
\Gamma(D(D_s)\rightarrow \mu^+\nu_\mu)/\Gamma(D(D_s)\rightarrow
\mu^+\tau_\tau)$ (and for some other heavy mesons $B^{\pm}$ and $B_c$ etc.) may
shed more light on physics responsible for the violation of the leptonic flavor
universality. The data of BES III are available to test the universality and
that of the future charm-tau factory will provide more accurate information
towards the aspect, in this work, we will discuss $R_{D(D_s)e\mu}$ and
$R_{D(D_s)\mu\tau}$ in all details and also briefly consider the cases for
$B^{\pm}$ and $B_c$.
