Analytical properties of $q$-metallic numbers

For an integer $n\geq 1$, consider the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and denote by $[\phi_n]_q$ its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a power series $[\phi_n]_q =\sum_{l=0}^{+\infty} \kappa_l(\phi_n) q^l$ around $q=0$, with integral coefficients. By using techniques from analytic combinatorics, we establish several properties of the sequence $( \kappa_l(\phi_n))_{l\geq 0}$ of Taylor coefficients: characterisation by recurrences or by differential equations, closed-form expressions when $n=1,2,3$, and asymptotics. We also present some remarkable identities induced by the action of the modular group $PSL(2,Z)$ and address, mainly through computer experimentations, the question of the logarithmic behaviour of the sequence $( \kappa_l(\phi_n))_{l\geq 0}$. A particular accent is put on the comparison between the $q$-deformation $[\phi_1]_q$ of the golden ratio and RNA secondary structures, the former being actually a signed version of the latter. By doing so, we would be pleased to bring the interest of combinatoricians to the newly discovered world of $q$-numbers.