Quantum $A_r$ Q-system solutions as q-multinomial series

We derive explicit expressions for the generating series of the fundamental solutions of the $A_r$ quantum $Q$-system of Ref. [P. Di Francesco and R. Kedem, arXiv:1006.4774 [math-ph]], expressed in terms of any admissible initial data. These involve products of quantum multinomial coefficients, coded by the initial data structure.