Quantum Periods from TBA and Continued Fractions

We consider quantum A-periods of local ${\mathbb P}^1 \times {\mathbb P}^1$, or equivalently the generating function of constant terms for powers of the Laurent polynomial $x + x^{-1} + y + y^{-1}$ where $x$ and $y$ are $q$-commuting variables. We derive a number of new representations for these ric