The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$

We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 %$ of the full textcolor{blue}{$O(T_F^2)$} and textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.