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{\bf Philippe Di Francesco }
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{\bf Quantum $A_r$ $Q$-System Solutions as q-Multinomial Series}
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We derive explicit expressions for the generating series of the fundamental 
solutions of the $A_r$ quantum $Q$-system of Di Francesco and Kedem [{\em 
Non-commutative integrability, paths and quasi-determinants}, Adv.\ in Math. 
\textbf{228(1)} (2011) 97--152], expressed in terms of any admissible initial data. 
These involve products of quantum multinomial coefficients, coded by the initial 
data structure.

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