Direct determination of the structure functions $F_L$, $F_S$ and $G$ from $F_2$ and $d{F}_2\mathrm{/}d{Q}^2$ to $O({\alpha }_s^2)$

We extend the results of Lappi {\em et al.}, Eur.~Phys.~J.~C {\bf 84}, 84 (2024), to show that it is possible to obtain expressions for the longitudinal, singlet and gluon structure functions $F_L$, $F_S$ and $G$ in deep inelastic scattering directly in terms of the measured functions $F_2$ and $d{F}_2\mathrm{/}\ln (Q^2)$ {\em modulo} non-singlet corrections expected to be small at very small $x$. The latter can be treated at low $x$ using existing quark distributions. Our results are presented consistently to $O({\alpha }_s^2)$, correcting and extending the mixed-order results of Lappi {\em et al.}.