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We introduce a q-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the q-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for P-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the q=1 case. We also show that certain properties enjoyed by the q-rationals are also satisfied by our higher versions.
Burcroff et al. (Tue,) studied this question.