Analytical properties of $q$-metallic numbers

TL;DR

This paper investigates the algebraic and combinatorial properties of $q$-deformed metallic numbers, including their series expansions, recurrences, identities, and connections to RNA structures, using analytic combinatorics and computer experiments.

Contribution

It provides new characterizations, closed-form formulas, and asymptotic behaviors of the $q$-metallic numbers, linking them to modular group actions and biological structures.

Findings

Established recurrence relations and differential equations for the coefficients.

Derived closed-form expressions for specific metallic numbers.

Explored the asymptotic and logarithmic behaviors of the coefficients.

Abstract

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.