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Han and Pedon’s Metallic Hankel Conjecture, Part 2

Abstract

For every n at least one, the shifted Hankel determinants of the q-metallic series are unbounded in absolute value at every shift at least n+3.

Theorem 1.1 (Unboundedness at all shifts at least n+3).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnbounded.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnbounded.result (✓ std3). ∎

Resolves. Problems/han-pedon-metallic-hankel-unbounded (proved) by D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnbounded.result.

Source. Repository-derived.

Acknowledgement. Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

Commentary.

For every integer n at least one, there exists an integral formal power series Phi with constant coefficient one satisfying q Phi^2 + ((1+q^n)(1-q)-q[n]_q)Phi = 1, where [n]q = 1+q+…+q^{n-1}. For every such Phi, every integer ell at least n+3 and every nonnegative integer M, some nonnegative integer j satisfies |Delta_j^{(ell)}| > M. Here Delta_j^{(ell)} is the determinant of the j by j matrix with entry [q^{ell+a+b}]Phi, with indices starting at zero and empty determinant one. At shift n+3 there is an explicit subsequence: for every positive integer k, Delta{Pk-1}^{(n+3)} = (-1)^{nk} 2k lambda, where P = 4 and lambda = 1 when n = 1, and P = 2n(n+1) and lambda = 2n+1 when n is at least two. The cofactor identity and dual-number transfer give this linear growth. The row at shift n+1 is bounded in absolute value by one. Exponential coefficient growth and the bounded-strip theorem then exclude boundedness at every later shift. This establishes part 2 of Conjecture E of Han and Pedon for all positive n.

References