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 Di Francesco and Kedem [ Non-commutative integrability, paths and quasi-determinants, Adv. in Math. 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.