Heavy-quark initiated charged-current deep-inelastic scattering coefficient functions through $\mathcal{O}(\alpha_s^2)$

We compute the coefficient functions for heavy-quark initiated charged-current deep-inelastic scattering through $\mathcal{O}(\alpha_s^2)$, retaining full mass dependence for a single heavy-quark flavor. The calculation employs a cut-based approach to isolate individual diagram contributions and to derive differential equations for the relevant master integrals, which are solved analytically in terms of generalized polylogarithms. The results are presented in the decoupling scheme for $n_L$ light active flavors, facilitating their direct implementation in variable-flavor number schemes such as ACOT and FONLL.