bibkey: sobolewski2026hankel authors: Bartosz Sobolewski, Maciej Ulas year: 2026 title: “Hankel determinants of weighted binary sums of digits” doi: 10.48550/arXiv.2607.09376 url: https://arxiv.org/abs/2607.09376v1 claim: “Recursions and closed forms for Hankel determinants of weighted binary digit sums; vanishing of the determinants at t = 2ζ on a structured index set and a conjectured exact description of the nonvanishing indices (Conjecture 5.7).” strata_touched:

• D5/S3/Combinatorics/DigitHankel/BinaryDigitHankel
• D5/S3/Combinatorics/DigitHankel/CyclotomicDigitHankel license: citation-only triage: anchor

Sobolewski and Ulas, Hankel determinants of weighted binary sums of digits

For u = Σ_j ε_j 2^j with binary digits ε_j and weights w_j, the weighted digit sum is s_w(u) = Σ_j ε_j w_j, and H_w(n) = det[s_w(i+j)]_{0≤i,j<n}. The paper derives a general recursion for H_w(n), closed forms along several index sequences for the ordinary digit sum, and, for w_j = t^j, studies the polynomial determinants H(n,t).

For t = 2ζ with ζ a primitive d-th root of unity, Theorem 5.4 shows that H(n,2ζ) vanishes on a structured set A_d, and Conjecture 5.7 states that it vanishes exactly there. For d = 2 the conjecture reads: H(n,−2) ≠ 0 if and only if n is one of n_k − 1, n_k, n_k + 1, where n_k = ⌈2^{k+2}/3⌉.

The module D5/S3/Combinatorics/DigitHankel/BinaryDigitHankel proves the case d = 2.

Verified locator

DOI: 10.48550/arXiv.2607.09376

URL: https://arxiv.org/abs/2607.09376v1

• Locator: Section 4, Theorem 4.4.
• Locator: Section 5.1, Theorem 5.4, Conjecture 5.7 and the sentence following it.