Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers

Guo-Niu Han, Emmanuel Pedon

Fix $n$ a positive integer. Take the $n$-th metallic number $\ddot{I}{•}_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\ddot{I}{•}_1$ is the golden number) and let $\hat{I}{¦}_n(q)$ be its $q$-deformation in the sense…