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A Cofactor Identity at Normal Hankel Indices

Abstract

Two orthogonal moment relations determine the shifted cofactor of a unit Hankel determinant.

Theorem 1.1 (The shifted cofactor from moment relations).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedCofactor.normal_cofactor

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedCofactor.normal_cofactor (✓ 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 be an integral formal power series, ell a nonnegative integer and m a positive integer. Let c and b be integer sequences and h an integer. Suppose c_m = 1, and b_1 = 0 when m = 1. For each t from zero through m-1, suppose the sum of c_r [q^{ell+r+t}]Phi over r from zero through m is zero. For each t from zero through m-2, suppose the sum of b_r [q^{ell+r+t}]Phi over r from zero through m-1 is zero, and suppose that the latter sum at t = m-1 equals h. If Delta_m^{(ell)} is an integer unit, then h Delta_{m-1}^{(ell+2)} = Delta_m^{(ell)}(c_1 b_0 - c_0 b_1). The unit condition means that Delta_m^{(ell)} is either one or minus one; no nonvanishing condition is imposed on the smaller determinant.

References