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Exponential Growth Bound for Metallic Coefficients

Abstract

The coefficients of every integral q-metallic solution admit a uniform exponential bound in their degree.

Theorem 1.1 (An exponential coefficient majorant).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedGrowth.coefficient_growth

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedGrowth.coefficient_growth (✓ 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 positive integer n, every integral formal power series Phi with constant coefficient one satisfying q Phi^2 + ((1+q^n)(1-q)-q[n]_q)Phi = 1, and every nonnegative integer m, the absolute value of [q^m]Phi is at most (4(n+5))^m. Here [n]_q = 1+q+…+q^{n-1}. The estimate includes m equal to zero and the golden case n equal to one.

References