Noncommutative Rational Series with Applications

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Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections (2018, background): “(c) The continuants enjoy many remarkable properties (some of which are listed in [4, 9, 17]).”
$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS (2018, background): “J8/3(q) = q ∣∣∣∣ [3]q q 2 1 [3]q ∣∣∣∣+ (1− q)[3]q = 1 + q + 2q 2 + q + 2q + q.”
Linear recurrence relations for cluster variables of affine quivers (2010, background): “ N. Suppose there exist m×m matrices A 0,...,As−1 over F such that for every j ∈ Nwe have vj+s = A 0vj + ... +As−1vj+s−1. Then each sequence (ai j)j∈N satisﬁes a linear recurrence. Proof. We refer to [5] for complete proofs of these fundamental facts. Let us record however, that if the two series ar”