Semileptonic decays of $D$- and $D_s$-mesons into orbitally excited states of strange and light mesons are studied in detail within the framework of the relativistic quark model based on the quasipotential approach and quantum chromodynamics. The form factors that parameterize the matrix elements of the weak current between meson states are calculated with consistent allowance for relativistic effects. Their dependence on the square of the transferred momentum is explicitly defined throughout the accessible kinematic region. Using the helical amplitude method, differential decay widths, branching fraction, various asymmetries and polarization characteristics, which can be measured experimentally, are calculated. The results obtained are in good agreement with the available experimental data.
$^2$Department of Quantum theory and High Energy Physics Physics, Faculty of Physics, M.V.Lomonosov Moscow State University