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=(n+√(n²+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 ℓ-shifts of Φ_n with 0≤ℓ≤ n+1. In addition, we formulate a new conjecture of quasi-polynomiality for higher shifts ℓ≥ 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.

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BibTeX RIS

BibTeX

@misc{han2026hankel,
  title = {{Hankel determinants for the (n+2)-shifted q-metallic numbers}},
  author = {Guo-Niu Han and Emmanuel Pedon},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36132},
  eprint = {2610.36132},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05841908v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hankel determinants for the (n+2)-shifted q-metallic numbers
AU  - Guo-Niu Han
AU  - Emmanuel Pedon
PY  - 2026
DA  - 2026/10/26
PB  - HAL
DO  - 10.48550/arxiv.2610.36132
UR  - https://hal.science/hal-05841908v1
AB  - Let Φ_n(q) denote the q-deformation (in the sense of Morier-Genoud & Ovsienko) of the metallic number ϕ_n=(n+√(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 ℓ-shifts of Φ_n with 0≤ℓ≤ n+1. In addition, we formulate a new conjecture of quasi-polynomiality for higher shifts ℓ≥ 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.
ER  -

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