Analytical properties of q-metallic numbers
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.
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BibTeX
@misc{pedon2026analytical,
title = {{Analytical properties of q-metallic numbers}},
author = {Emmanuel Pedon},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2604.19898},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05602801v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytical properties of q-metallic numbers
AU - Emmanuel Pedon
PY - 2026
DA - 2026/09/24
PB - HAL
UR - https://hal.science/hal-05602801v2
AB - 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.
ER -