Hankel Determinants of Geometric Moments and Rational Lambert Values

Will Cook

Short paper

Hankel Determinants of Geometric Moments and Rational Lambert Values

Erdős #1049 By Will Cook September 2026 Authorship, AI use and citation

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Summary. Positive geometric moments and a uniform partition-sum bound give the fixed-base asymptotic of Zudilin’s normalised Hankel determinants, including its constant and power correction. A separate calculation of degrees after cancellation in his 2004 forms extends the rational-base irrationality region. It includes every positive integral power of $31 / 4$ and excludes $3 / 2$ .

This paper covers the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at $3 / 2$ .

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open.

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In this paper

The first nonzero formal term

The order of a nonzero formal series is the least exponent with nonzero coefficient. We now fix $N$ and determine the first term in the formal $q$ -expansion. The shifted term in (8) then has an extra factor of $q$ . More generally, every nonempty partition has positive displacement exponent. Thus a single tuple determines the first formal term, although all partitions were needed for the fixed-base constant.

Theorem 3.2 (the first nonzero term of the Hankel determinant). For every $N \geq 1$ ,

{ord}_{q} V_{N}^{*} = \frac{N (N - 1) (2 N - 1)}{6},

and the coefficient of the first nonzero monomial is

[q^{N (N - 1) (2 N - 1) / 6}] V_{N}^{*} = \frac{(N!)^{2} (N + 1)!}{2^{N}} .

Proof. At $q = 0$ , the $t = 0$ term of $G_{q} (w)$ is $(1 - w^{2}) / (1 - w)^{5}$ , while the remaining terms sum to $w / (1 - w)^{4}$ . Thus

G_{0} (w) = \frac{1 + 2 w}{(1 - w)^{4}}, a_{k} (0) = c_{k} > 0.

We compute modulo $q^{M + 1}$ by truncating each moment sum in (12) at $k = M$ . Since all omitted terms are divisible by $q^{M + 1}$ , multilinearity leaves the determinant unchanged modulo that power. Applying finite Cauchy–Binet and then allowing $M$ to increase, we obtain the coefficientwise identity

V_{N}^{*} = \sum_{k_{0} < \dots < k_{N - 1}} (\prod_{i < N} a_{k_{i}} (q) q^{k_{i}}) \prod_{i < j < N} (q^{k_{i}} - q^{k_{j}})^{2} .

The order of the summand indexed by $(k_{i})$ is $\sum_{i < N} (2 N - 1 - 2 i) k_{i}$ . Because $k_{i} \geq i$ and all the weights $2 N - 1 - 2 i$ are positive, we obtain the unique minimum at $k_{i} = i$ , with order and coefficient

\sum_{i < N} (2 N - 1 - 2 i) i = B_{N}, \prod_{i < N} c_{i} = C_{N} .

No other tuple has this order, so no coefficient from another tuple can cancel it. This argument needs the positive constant terms $a_{k} (0) = c_{k}$ , not nonnegativity of all their formal coefficients. For $N = 2$ , the pair $(0, 1)$ contributes $6 q + O (q^{2})$ , and every other increasing pair has order at least two. ◻

This proves equality in Zudilin’s bound [5]. An alternative argument, which computes the first nonzero term of every transformed row, is given in the companion paper.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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