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Dilated Hankel determinants

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Pith's one-line read For many classical sequences the dilated Hankel determinant factors into a simple product, even when the ordinary Hankel determinant does not.

desk verdict Solid catalogue of product formulas for dilated Hankel minors, six reusable methods with clear scope limits, and a clean settlement of the Chapoton–Han root conjecture. read the letter →

arxiv 2607.08279 v1 pith:HATJ4RVZ submitted 2026-07-09 math.CO math.NT

Guo-Niu Han This is my paper · ORCID
classification math.COmath.NT MSC 11C2015A1533C4505A1011A5511B6811B83
keywords HankeldeterminantdilatedorthogonalpolynomialsJacobicontinuedfractionCatalannumbersEulerSpringerbiorthogonalreduction
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces the dilated Hankel determinant of a sequence—the minor of the infinite Hankel matrix that keeps the even-indexed rows and the first n columns—and shows that this object admits closed product formulas for a surprisingly wide range of classical sequences. The same sequences often have ordinary Hankel determinants that either lack product formulas or require the full machinery of Jacobi continued fractions; here there is no single universal tool, so the author develops six methods (Vandermonde reduction, biorthogonal reduction, one-functional reduction, divisor method, contiguous relations, and a rank-one matrix-determinant lemma) and applies them family by family. The resulting evaluations cover factorials, Catalan and central binomial numbers, involutions, Euler and secant families, Springer numbers and their elliptic and derivative deformations, a reciprocal-sine generating function, Bessel analogues, and algebraic families. As a concrete application the paper settles a prior conjecture on the roots of the Poupard and Kreweras polynomials by identifying that conjecture, up to an explicit power of two, with one of the newly evaluated dilated determinants.

What carries the argument

Six evaluation methods for the dilated determinant, of which the dominant one is the biorthogonal reduction: the sequence is split into even and odd moment functionals, the dilated matrix is reduced to a determinant of connection coefficients between their orthogonal polynomials, and that determinant collapses to a product whenever the two functionals form a one-parameter classical pair whose connection kernel is evaluable.

What would settle it

Compute the dilated Hankel determinant of a sequence whose even and odd moment functionals are classical but differ in two or more parameters (for example a genuine two-parameter Wilson pair) and check whether the resulting integer sequence still factors into small linear terms; the appearance of large sporadic primes already at moderate order would confirm the claimed collapse boundary.

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Extended reading notes

Core claim

For a broad class of classical sequences the dilated Hankel determinant ¨H_n(a)=det(a_{2i+j}) admits a simple product evaluation, even though no universal continued-fraction formula exists for it and the class of sequences with known product formulas is larger for the ordinary Hankel determinant. The evaluations are obtained by six methods developed in the paper and include, among others, the factorial, Catalan, Euler/secant, Springer, reciprocal-sine and Bessel families; they also settle a conjecture on the roots of the Poupard and Kreweras polynomials.

Load-bearing premise

The main product formulas hold only when the even and odd moment functionals differ by a single classical parameter so that their connection coefficients reduce to a single hypergeometric term whose kernel can be evaluated; outside that structural window the paper itself expects no product formula.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper defines the dilated Hankel determinant ¨H_n(a)=det(a_{2i+j})_{0≤i,j≤n−1} and proves that it admits simple product evaluations for a wide range of classical sequences (factorials, Catalan and central binomials, Euler/secant families and shifts, involutions/Gaussian, Springer and elliptic deformations, reciprocal-sine, algebraic families, Bessel analogues). Six methods are developed (Vandermonde M1, biorthogonal M2, one-functional M3, divisor/Cauchy–Binet M4, contiguous M5, rank-one/matrix-determinant lemma M6), four of general scope. As an application, Conjecture 5.4 of Chapoton–Han on the roots of the Poupard and Kreweras polynomials is reduced (after a change of basis and an explicit power of two) to the dilated determinant of (1+x)/cos x and settled by the s=0 evaluation of the secant family.

Significance. The work supplies a large collection of previously unrecorded closed forms for a natural Hankel minor, together with a systematic toolkit that has no single universal counterpart to the Heilermann–Stieltjes formula. The reciprocal-sine evaluation (via a new Catalan determinant proved by condensation) and the complete settlement of the Chapoton–Han conjecture are particularly valuable. Strengths include fully written proofs, explicit lemmas, and the transparent statement of the structural hypothesis under which the biorthogonal reduction collapses. The results are of clear interest to combinatorialists and special-function theorists working with Hankel determinants, continued fractions and orthogonal polynomials.

minor comments (4)
  1. The manuscript is very long (30 sections). A short “roadmap” paragraph at the end of the introduction, listing which families use which of M1–M6, would help the reader navigate.
  2. Section 3.5 Table 1 is useful; ensuring that every later section explicitly tags the method(s) used (as promised) would improve cross-referencing.
  3. A few of the longer inductive verifications (e.g., connection-coefficient recurrences) are described as “elementary polynomial identities, routine to verify.” Adding a short SageMath or computer-algebra appendix (or a public repository link) would make those steps fully reproducible.
  4. Notation for the various shifts ¨H^{(1)}_n, ¨H^{(2)}_n is introduced gradually; a single summary table of all variants would reduce the cognitive load.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: product evaluations are derived from independent orthogonal-polynomial, Vandermonde, divisor, contiguous, and matrix-determinant arguments; the Chapoton–Han self-citation only names the conjecture being settled.

full rationale

The paper’s load-bearing claims are explicit product formulas for ¨H_n of named classical families, obtained by six methods (M1–M6) developed in Section 3 and applied family-by-family. Each method reduces the dilated determinant to standard objects (Vandermonde factors, connection-coefficient determinants of one-parameter classical pairs, Hermite triangularisation, degree-and-zero divisor bookkeeping, contiguous column operations, or the matrix-determinant lemma) whose evaluations are proved inside the paper by recurrence verification, condensation, or elementary polynomial identities. Specialisations that fix multiplicative constants (e.g. s=0 for the secant family after the degree/divisibility argument of Theorem 12.1, or s=2 for the algebraic family after the contiguous relation of Theorem 20.1) invoke earlier independent evaluations (Proposition 11.1, Corollary 4.4), not the target formula itself. The sole self-citation of overlapping authorship is [4] (Chapoton–Han), used only to identify Conjecture 5.4; Theorem 26.3 reduces that conjecture, after a change of basis and an explicit power of two, to the already-proved product of Proposition 11.1. No step equates a claimed prediction with a fitted input, imports an unverified uniqueness theorem from the same authors as an external fact, or renames a known empirical pattern as a derivation. The derivation chain is therefore self-contained against its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The work is pure formal algebra over fields/power series. It imports standard orthogonal-polynomial and continued-fraction machinery, classical determinant identities, and known generating functions for Euler, Springer, Catalan, Bessel, etc. No numerical free parameters are fitted. Invented entities are only the named methods and the dilated determinant itself, which are definitions rather than physical postulates.

assumptions (6)
  • standard math Quasi-definiteness of moment functionals (Hn ≠ 0) so that monic orthogonal polynomials and three-term recurrences exist (Favard).
    Used throughout M2/M3 reductions (Section 2–3); classical hypothesis for the moment problem and OP theory.
  • standard math Desnanot–Jacobi (Dodgson) condensation identity for minors.
    Finishing step for the Catalan determinant (Theorem 18.8) and the Bessel connection determinant (Section 23).
  • standard math Even/odd contractions of Stieltjes S-fractions to Jacobi J-fractions and Christoffel transforms (Stieltjes, Flajolet).
    Supplies recurrence coefficients and norms for Euler, secant, Springer-type families (Section 2.6–2.7).
  • standard math Dual Jacobi–Trudi / hook-content evaluation of rectangular (and near-rectangular) Schur specializations to binomial determinants.
    Lemma 7.4 and related binomial determinants in the Euler and single-shift sections.
  • domain assumption Classical integral/weight representations and recurrence data for Wilson, continuous dual Hahn, Hermite, and Jacobi elliptic Fourier/nome expansions.
    Used to identify even/odd functionals for reciprocal-sine, secant, Gaussian, and elliptic Euler families (Sections 5, 11, 18, 19); taken from standard references (Askey scheme, Whittaker–Watson).
  • domain assumption Collapse of connection coefficients to a single hypergeometric term when two classical functionals differ in one parameter.
    Stated as the working mechanism of M2 (end of Section 3.2); standard for one-parameter classical pairs but is the scope boundary of the product formulas.
invented entities (2)
  • Dilated Hankel determinant ¨Hn(a)=det(a_{2i+j}) independent evidence
    purpose: Primary object of study; even-row minor of the infinite Hankel matrix.
    Definitional; not a physical postulate. Independent interest via Gram determinants of even/odd bases and application to Poupard–Kreweras roots.
  • Six methods M1–M6 (Vandermonde, biorthogonal, one-functional, divisor/Cauchy–Binet, contiguous, rank-one/matrix-determinant lemma) independent evidence
    purpose: Organise all evaluations in the absence of a universal Heilermann-type formula for the dilated case.
    Methodological packaging of classical techniques plus specialised tools; each is proved as a lemma before use.

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Pith. "Pith review of Dilated Hankel determinants." pith.science (2026). https://pith.science/paper/HATJ4RVZ

@misc{pith2026260708279,
  author       = {Pith},
  title        = {Pith review of: Dilated Hankel determinants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HATJ4RVZ}},
  note         = {Machine review of arXiv:2607.08279}
}
abstract

For a sequence $\mathbf a=(a_0,a_1,\dots)$ we define its dilated Hankel determinant $\ddot{H}_n(\mathbf a)=\det(a_{2i+j})_{0\le i,j\le n-1}$, the minor of the infinite Hankel matrix $(a_{i+j})$ formed from the even-indexed rows and the first $n$ columns. We prove that, for a broad class of sequences, $\ddot{H}_n$ admits a remarkably simple product evaluation. This mirrors the behaviour of the classical Hankel determinant $H_n$, but with two key distinctions: the class of sequences for which such formulas are known is far larger in the classical case; and, whereas $H_n$ enjoys a single universal evaluation -- the Heilermann formula via the Jacobi continued fraction -- no analogous general method exists for the dilated determinant, which is therefore considerably more challenging. Our evaluations instead rest on six methods developed here, four of general scope and two of a more specialised nature. The cases treated include the factorial numbers, the Catalan and central binomial coefficients; the Euler numbers and a one-parameter secant family; the involution numbers; the Springer numbers along with elliptic and derivative deformations; the reciprocal-sine function, whose evaluation rests on a new Catalan determinant proved by condensation; a Bessel analogue of the Euler numbers; and a multiplicative Bessel family. As an application, we settle a conjecture of Chapoton and the author on the roots of the Poupard and Kreweras polynomials.

Figures

Figures reproduced from arXiv: 2607.08279 by the authors.

Figure 1
Figure 1. The classical Hn = H({0, 1, 2, 3}; {0, 1, 2, 3}): equally spaced, sym￾metric nested arches. The dilated determinant is the same picture with the sources pulled apart. In H¨ n the top starting points move from 0, 1, . . . , n−1 to 0, 2, . . . , 2n−2 while the sinks are unchanged ( [PITH_FULL_IMAGE:figures/full_fig_p095_1.png] view at source ↗
Figure 2
Figure 2. The dilated H¨ n = H({0, 2, 4, 6}; {0, 1, 2, 3}): rows dilated, arches skewed, telescoping lost. 28.3. The even–odd splitting: the biorthogonal picture. There is nonetheless a way to keep H¨ n subtraction-free, by splitting it into two path systems — one carried by the even moments, one by the odd moments. This is the combinatorial face of the biorthogonal reduction M2. Write p = ⌈n/2⌉ and q = ⌊n/2⌋. In H¨ n = (a2i+… view at source ↗
Figure 3
Figure 3. The even–odd splitting of H¨ n (n = 6, generic): even system I = {0, 2, 3} above, odd system I c = {1, 4, 5} mirrored below. 29. Classical Hankel determinants Alongside the dilated determinant H¨ n = det(a2i+j )0≤i,j≤n−1, each moment sequence produced in this paper also has a classical Hankel determinant Hn(µ) = det µi+j  0≤i,j≤n−1 . The evaluation of such determinants is a classical and much-studied subject; for s… view at source ↗

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