  Motivated by the disagreement between the experimental data and lattice
calculations on the decay constant of the $D_{s}$ meson, we investigate
leptoquark (LQ) contributions to the purely leptonic decays of a pseudoscalar
(P). We concentrate on the LQs which only couple to the second-generation
quarks before the electroweak symmetry breaking and we discuss in detail how
flavor symmetry breaking effects are brought into the extension of the standard
model after the spontaneous symmetry breaking. We find that the hermiticity of
fermion mass matrices can not only reproduce the correct
Cabibbo-Kobayashi-Maskawa and Maki-Nakagawa-Sakata matrices, but also reduce
the number of independent flavor mixing matrices and lead to
$V^{R}_{f}=V^{L}_{f}$ with $L(R)$ denoting the chirality of the f-type fermion.
Accordingly, it is found that the decays $D_{s,d}\to \ell^{+} \nu$, $B^{+}\to
\tau^{+} \nu$ and $B_c\to \ell^{+} \nu$ have a strong correlation in
parameters. We predict that the decay constant of the $B_c$ meson calculated by
the lattice could be less than the experimental data by 23%. Intriguingly, the
resultant upper limits of branching ratios for $D\to \mu^{+} \mu^{-}$ and
$\tau\to \mu (\pi^0, \eta, \eta', \rho, \omega)$ are found to be around $
5.1\times 10^{-7}$ and $(2.6, 1.5, 0.6, 7.4, 4.8)\times 10^{-8}$, which are
below and close to the current experimental upper bounds, respectively.
