  We calculate the massive operator matrix element $A_{gq}^{(3)}(N)$ to 3-loop
order in Quantum Chromodynamics at general values of the Mellin variable $N$.
This is the first complete transition function needed in the variable flavor
number scheme obtained at $O(\alpha_s^3)$. A first independent recalculation is
performed for the contributions $\propto N_F$ of the 3-loop anomalous dimension
$\gamma_{gq}^{(2)}(N)$.
