The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O(\alpha_s^3)$

We calculate the polarized massive operator matrix element $A_{gq}^{(3)}(N)$ to 3-loop order in Quantum Chromodynamics analytically at general values of the Mellin variable $N$ both in the single- and double-mass case in the Larin scheme. It is a transition function required in the variable flavor number scheme at $O(\alpha_s^3)$. We also present the results in momentum fraction space