Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

The Large-Shift Metallic Hankel Conjecture

Abstract

For every positive metallic parameter, each shifted Hankel determinant sequence at shifts at least n+3 is unbounded in absolute value.

Definition 1.1 (Unboundedness at every large shift).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedDefs.claim

Formalization. D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedDefs.claim (✓ std3).

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, there is a nonnegative integer j such that the absolute value of Delta_j^{(ell)} exceeds M. Here Delta_j^{(ell)} is the determinant of the j by j matrix with entry [q^{ell+a+b}]Phi, with row and column indices starting at zero and empty determinant one. This is part 2 of Conjecture E of Han and Pedon, with ell as the varying shift.

References