Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Evaluations on the Threshold Triples

Abstract

Reflection induction evaluates the binary digit Hankel determinants at each center and its two adjacent indices.

Theorem 1.1 (The two adjacent determinant families).

Lean statement: D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelNonzero.endpoint_evaluations

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelNonzero.endpoint_evaluations (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bartosz Sobolewski, Maciej Ulas (2026). Hankel determinants of weighted binary sums of digits. DOI: 10.48550/arXiv.2607.09376. URL: https://arxiv.org/abs/2607.09376v1.

Commentary.

For every integer k at least two, put n_k = ceil(2^(k+2)/3). Then H(n_k-1,-2) = -2^((k+1)n_k - floor(3k/2) - 4) and H(n_k+1,-2) = 2^((k+1)n_k + floor(k/2) + 2 - (k mod 2)). Moreover E(n_k-1,-2) = 0 and E(n_k+1,-2) = 0. The initial values at k = 2 and the paired reflection recurrences give a simultaneous induction: reflection interchanges the two adjacent families, while the vanishing of E removes the additive term in the recurrence for H.

Theorem 1.2 (The determinant families at the centers).

Lean statement: D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelNonzero.center_evaluations

Proof. Machine-checked in Lean as D5/S3/Combinatorics/DigitHankel/BinaryDigitHankelNonzero.center_evaluations (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bartosz Sobolewski, Maciej Ulas (2026). Hankel determinants of weighted binary sums of digits. DOI: 10.48550/arXiv.2607.09376. URL: https://arxiv.org/abs/2607.09376v1.

Commentary.

For every nonnegative integer k, put n_k = ceil(2^(k+2)/3). Then E(n_k,-2) = (-1)^k 2^((k+1)n_k - floor((k+1)/2)) and 12 H(n_k,-2) = -(2^(k+1) + (-1)^k) E(n_k,-2). Reflection sends the center n_k to the preceding center for k at least two. The initial values at k = 0 and k = 1 and simultaneous induction give the two identities.

References