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The Shifted Hankel Desnanot-Jacobi Identity

Abstract

Shifted Hankel determinants of any integral formal power series satisfy the Desnanot-Jacobi identity, including at vanishing determinants.

Theorem 1.1 (Adjacent shifts and determinant sizes).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedJacobi.hankel_jacobi

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedJacobi.hankel_jacobi (✓ 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 integral formal power series Phi and all nonnegative integers ell and j, write Delta_r^{(s)} for the determinant with entries [q^{s+a+b}]Phi and size r. Then (Delta_{j+1}^{(ell+1)})^2 = Delta_{j+1}^{(ell)} Delta_{j+1}^{(ell+2)} - Delta_{j+2}^{(ell)} Delta_j^{(ell+2)}. The empty determinant is one. No determinant is assumed nonzero, and the identity holds also for j equal to zero.

References