  We provide compact formulas for the heavy quarks structure functions
$F^{q\bar{q}}_{2}(x, Q^{2})$ and $F^{q\bar{q}}_{L}(x, Q^{2})$, with $q = c, b$
and $t$, in $e^{-}p$ interaction with respect to the behavior of the gluon
density at the next-to-leading order (NLO) approximation and present ratios $R
^{q\bar{q}} = F^{q\bar{q}}_{L} /F^{q\bar{q}}_{2}$ for these quarks. To do this,
we obtain a fitted form of the heavy quark coefficient functions for
deep-inelastic lepton-hadron scattering at a wide range of $Q^{2}$ values. The
obtained numerical results are compared with experimental data from HERA and
with the results from the KLN model and the MSTW2008 predictions.
