bibkey: rosenzweigstanfill2026loglaplacians authors: Bart Rosenzweig and Jonathan Stanfill year: 2026 title: “On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians” doi: 10.48550/arXiv.2606.04225 url: https://arxiv.org/abs/2606.04225v1 claim: “Open Problem 1.6(iv) asks whether expression (1.13) vanishes for every alpha in (0, pi) whenever m is nonnegative and even.” strata_touched:

• D5/S3/ArithSums/LogLaplacianEvenResidueVanishing license: citation-only triage: anchor

Rosenzweig–Stanfill logarithmic Laplacians

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• DOI: 10.48550/arXiv.2606.04225
• URL: https://arxiv.org/abs/2606.04225v1
• Version: arXiv:2606.04225v1.
• Open Problem 1.3, printed page 2: show that $p_{2m,S_1}(1/2-m)\ne 0$ for every positive natural $m$, with $S_1$ defined by equation (1.8).

Source statement

Open Problem 1.6, printed page 4:

Consider the notation of Theorem 1.5. Then the following are conjectured to be true:

(iv) For every α ∈ (0, π), (1.13) is equal to zero whenever m ≥ 0 is even.

Definition 1.1, printed page 2:

Given a sequence of numbers, S, indexed over a set J ⊇ N, we define

$$p_{j,S}(t):=\sum _{k=0}^j\frac{(-1{)}^k}{k!}{\hat{B}}_{j,k}(s_1,\dots ,s_{j-k+1})t^k,j\in {\mathbb{N}}_0.$$

Theorem 1.5, printed page 4:

where the sequence S₂ satisfies sₖ⁽²⁾ = −Bₖ/k, k ∈ N.

The paper fixes $B_1=-1/2$. Equations (1.20)–(1.21), printed page 6, define the partial ordinary Bell polynomials by a multinomial sum over nonnegative profiles with total count $k$ and weighted count $n$. The formal encoding uses these formulas, the recursive primary $c$-array of (1.14), its separate value at $\alpha =\pi /2$, and the finite $d$ and $b$ arrays. The zero entry of the positive-index Bell sequence is unused. The expression (1.13) is $2e^{{\gamma }_{E}m}/\pi$ times the two-sum bracket; this nonzero factor does not affect vanishing.

The settlement concerns clause (iv). Clauses (i)–(iii), the analytic construction of the fundamental solution, and analytic convergence of its power series are separate statements.