The transition matrix element Agq(N)Agq(N) of the variable flavor number scheme at O(αs3)

We calculate the massive unpolarized operator matrix element Agq(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(αs3). A first independent recalculation is performed for the contributions ∝NF∝NF of the 3-loop anomalous dimension γgq(2)(N).