bibkey: han2025hankel authors: Guo-Niu Han, Emmanuel Pedon year: 2025 title: “Hankel continued fractions and Hankel determinants for q-deformed metallic numbers” doi: 10.48550/arXiv.2502.05993 url: https://arxiv.org/abs/2502.05993v2 claim: “H-fraction expansions of the q-metallic numbers and their shifts up to n+1, periodicity, values, Gale-Robinson recurrences and contiguity relations of their shifted Hankel determinants; Conjecture E on the shifts n+2 and beyond.” strata_touched:

• D5/S3/Combinatorics/MetallicHankel/MetallicHankel
• D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnbounded license: citation-only triage: anchor

Han and Pedon, Hankel determinants of q-deformed metallic numbers

The q-metallic number Φ_n(q) is the power series with constant term 1 satisfying q Φ² + ((1 + qⁿ)(1 − q) − q[n]_q) Φ = 1, a q-deformation of the metallic ratio in the sense of Morier-Genoud and Ovsienko. The paper computes the Hankel continued fraction of Φ_n and of its shifts Φ_n^{(ℓ)} for ℓ ≤ n + 1 (Theorems A and 7.3), and proves that the shifted Hankel determinants Δ^{(ℓ)}, 0 ≤ ℓ ≤ n + 1, take values in {−1, 0, 1}, are 2n(n+1)-periodic for even n and antiperiodic for odd n (Theorem B), satisfy a Gale–Robinson recurrence (Theorem C) and contiguity relations (Theorem D). Conjecture E states that Δ^{(n+2)} has the same (anti)periodicity with values in {−2, …, 2} (part 1) and that Δ^{(ℓ)} is unbounded for ℓ ≥ n + 3 (part 2).

The module D5/S3/Combinatorics/MetallicHankel/MetallicHankel proves part 1 of Conjecture E for every n ≥ 2.

Verified locator

DOI: 10.48550/arXiv.2502.05993

URL: https://arxiv.org/abs/2502.05993v2

• Locator: Section 1, equations (1.1), (1.2), Theorems A–D and Conjecture E.
• Locator: Section 2, equation (2.5); Section 7, Theorem 7.3.