Unimodality of q-Fibonomial polynomials and related quotients

We prove that every q-Fibonomial polynomial is symmetric and unimodal, settling a conjecture of Bergeron, Ceballos, and Küstner. Its coefficients are strictly increasing in the first half whenever both parameters are at least 9. The proof expresses adjacent coefficient differences as a principal contribution governed by a convolution of interval indicators and a sum of contributions from other roots of unity. Fibonacci addition identities give a uniform lower bound for the former, while strong divisibility and root-of-unity moment estimates control the latter. For general quotients of q-integers with fixed denominators, we prove eventual unimodality under strict balance of the limiting numerator lengths and a pole multiplicity gap. We also establish a growth-rate obstruction for strong divisibility sequences, prove unimodality for strict divisibility chains, and classify the positive normalized coprime second-order recurrences whose quotients are unimodal for all parameters.