Proof of Barry's Four Hankel Determinant Conjectures

Feihu Liu; Ying Wang; Zihao Zhang

Proof of Barry's Four Hankel Determinant Conjectures

Combinatorics 2026-07-21 v1

Authors: Feihu Liu , Ying Wang , Zihao Zhang

Abstract

Barry introduced a central transform of integer sequences and proposed four conjectures concerning the Hankel transforms of central transform of four rational families. We prove these four conjectures. The proofs are unified within a common algebraic framework: we interpret the Hankel determinants as Gram determinants and use a basis of shifted monic Chebyshev polynomials to reveal the finite-band structure of the associated Gram matrices.

Keywords

Hankel determinant Hankel transform central transform Gram determinant Chebyshev polynomials integer sequences

Cite

@article{arxiv.2607.18644,
  title  = {Proof of Barry's Four Hankel Determinant Conjectures},
  author = {Feihu Liu and Ying Wang and Zihao Zhang},
  journal= {arXiv preprint arXiv:2607.18644},
  year   = {2026}
}

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33 pages