Hankel determinants for the $(n+2)$-shifted $q$-metallic numbers

Abstract

Let $Φ_n(q)$ denote the $q$-deformation (in the sense of Morier-Genoud & Ovsienko) of the metallic number $ϕ_n=\frac{n+\sqrt{n^2+4}}{2}$. This is a power series with integer coefficients. We prove that the Hankel continued fraction of the $(n+2)$-shifted series $Φ_n^{(n+2)}$ is eventually periodic, and that the corresponding Hankel determinant sequence $Δ^{(n+2)}$ is $2n(n+1)$-periodic if $n$ is even, $2n(n+1)$-antiperiodic if $n$ is odd, and that it takes only the values $-2,-1,0,1,2$. This confirms part of a former conjecture of ours and complements similar results we obtained for all $\ell$-shifts of $Φ_n$ with $0\le\ell\le n+1$. In addition, we formulate a new conjecture of quasi-polynomiality for higher shifts $\ell\ge n+3$. The determinant values arise as telescoping products of the partial numerators of the Hankel continued fraction. Most of the technically involved and lengthy verifications are carried out by a symbolic computation program, whose code is publicly available on GitHub for independent verification.