Heavy quarks structure functions $F_2^{q\bar{q}}(x,Q^2)$ and $F_L^{q\bar{q}}(x,Q^2)$ at the NLO approximation

We provide compact formulas for the heavy quarks structure functions $F_2^{q\bar{q}}(x,Q^2)$ and $F_L^{q\bar{q}}(x,Q^2)$, 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 $R^{q\bar{q}}=F_L^{q\bar{q}}\mathrm{/}{F}_2^{q\bar{q}}$ 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.