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Propagation of Hankel Periodicity Across Zero Intervals

Abstract

Periodic monic moment relations propagate a signed period and a five-value bound to every determinant size.

Theorem 1.1 (A signed period for the next Hankel shift).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelPeriod.propagate

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelPeriod.propagate (✓ 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.

Let Phi(q) = sum_{r >= 0} f_r q^r be an integral formal power series, and let ell,n,L,P be nonnegative integers with P even. Suppose s_0 = 0 and s_{p+1} = s_p+k_p+1. Assume s_{p+L} = s_p+P, k_{p+L} = k_p, h_{p+L} = h_p and c_{p+L,0} = c_{p,0} for every p, with h_p in {-1,1}, c_{p,0} in {-1,0,1,2} and c_{p,s_p} = 1. The moment sum sum_{r=0}^{s_p} c_{p,r} f_{ell+r+t} must vanish for t < s_{p+1}-1 and equal h_p at t = s_{p+1}-1. Suppose also that, for every p, the product of (-1)^{k_{p+i}(k_{p+i}+1)/2} h_{p+i}^{k_{p+i}+1} over 0 at most i and i less than L is (-1)^n. Then for every nonnegative j, Delta_{j+P}^{(ell+1)} = (-1)^n Delta_j^{(ell+1)}, and Delta_j^{(ell+1)} belongs to {-2,-1,0,1,2}. The conclusion includes determinant sizes between successive s_p, where the preceding-shift Hankel determinants vanish.

References