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We derive explicit expressions for the generating series of the fundamentalsolutions 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 initialdata. These involve products of quantum multinomial coefficients, coded by theinitial data structure.

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