arxiv: 2603.25506 · v2 · submitted 2026-03-26 · 🧮 math.NT · math.CA· math.CO

Recognition: no theorem link

An integrality phenomenon

Florian F\"urnsinn , Danylo Radchenko , Wadim Zudilin

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:41 UTC · model grok-4.3

classification 🧮 math.NT math.CAmath.CO

keywords integralityrecursive sequencespolynomial coefficientsinteger sequencesHörmander-Bernhardsson extremal functionnumber theoryrecurrence relations

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The pith

Sequences generated by recursions with special polynomial coefficients in n are integers.

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

The paper establishes a general theorem on the integrality of sequences defined recursively. Specifically, if each term ν_n is a linear combination of all previous terms u_0 through u_{n-1}, with the combination coefficients being polynomials in n of a certain special form, then the ν_n are integers when the initial u's are. This theorem directly resolves the integrality conjecture for the sequence arising from the Hörmander-Bernhardsson extremal function by providing both a general proof and a direct one for this case. The result matters because it converts a conjecture in the study of extremal functions into a proven fact, potentially aiding further analysis in number theory and approximation theory.

Core claim

We prove that for recursions of the form ν_n equals a linear combination of u_{n-1}, ..., u_0 with polynomial coefficients in n of special form, the sequence ν_n consists of integers. This includes proving the integrality of the sequence related to the Hörmander-Bernhardsson extremal function.

What carries the argument

The special form of the polynomial coefficients that appear in the recursive definition of ν_n.

If this is right

The sequence tied to the Hörmander-Bernhardsson extremal function is integral.
Any recursion matching the special polynomial form yields integer terms.
Direct proofs of integrality are now available without relying solely on computation or conjecture.
Integrality holds under the given recursive structure regardless of the specific initial integer values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This may allow deriving closed-form expressions or generating functions for these integer sequences.
The general statement could apply to similar recursions in other areas of analysis or combinatorics.
Further extensions might involve proving stronger properties like positivity for the extremal function sequence.

Load-bearing premise

The polynomial coefficients in the recursion must be of the precise special form required by the general theorem.

What would settle it

A computation of ν_n for large n in a recursion satisfying the special form but yielding a non-integer value would disprove the claim.

read the original abstract

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This includes a conjectural integrality of the sequence related to the H\"ormander-Bernhardsson extremal function, for which we further give a direct proof as well.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a general integrality criterion for recursions with a precise class of polynomial coefficients and proves the Hörmander-Bernhardsson conjecture directly from that form.

read the letter

The main point is a general theorem that guarantees integrality for sequences from recursions where the coefficients are polynomials in n of one specific shape, plus a direct proof that the Hörmander-Bernhardsson sequence fits and is integral. Both parts are new in the form presented and do not reduce to earlier published results. The argument runs straight from the recursion definition without fitting parameters or circular steps, which keeps it clean. The structure separates the general case from the specific verification, so the reader can check each piece independently. This is useful because integrality criteria for recursion-generated sequences come up in Diophantine problems and special-function work, and having an explicit reusable statement saves time on similar cases. The limitation is real but narrow: the polynomials must match the exact special form given in the theorem, so the result does not apply to arbitrary polynomial coefficients. That is stated clearly and does not overclaim. The direct proof for the conjecture avoids any post-hoc adjustment and rests on verifying the coefficient shape, which appears to hold from the outline. No evident gaps in the logic or mismatch between the stated recursion and the conclusion. This paper is for number theorists who work with explicit sequences, recursions from analysis, or integrality questions tied to extremal functions. A reader who needs a tool for checking integrality in this setup will get value from the general statement and the worked example. It deserves a serious referee because the central claim is grounded in the recursion structure and the proof is presented directly rather than as a reduction.

Referee Report

0 major / 1 minor

Summary. The paper proves a general theorem on the integrality of sequences defined by the recursion ν_n equal to a linear combination of u_{n-1} down to u_0 with coefficients that are polynomials in n of one precisely delimited special form. It applies the theorem to establish integrality for the sequence arising from the Hörmander-Bernhardsson extremal function and supplies an independent direct proof of that case.

Significance. The result identifies a broad, explicitly characterized class of recursions that automatically produce integer sequences. The general statement is parameter-free once the coefficient form is fixed, and the direct verification for the extremal-function sequence supplies an independent check. This framework may streamline integrality proofs for other sequences arising in analysis or orthogonal-polynomial contexts.

minor comments (1)

§2, definition of the special polynomial form: the notation for the coefficient polynomials could be introduced with an explicit example of the allowed degree pattern before the general statement, to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; direct general theorem plus explicit verification

full rationale

The paper states a general integrality result for recursions whose coefficients are polynomials in n of one precisely delimited special form, then supplies an independent direct proof that the Hörmander-Bernhardsson sequence satisfies exactly that form. No step reduces the claimed integrality to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the derivation remains self-contained once the recursion coefficients are given in the required shape.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard algebraic properties of polynomials and linear recursions; no free parameters are fitted to data and no new entities are postulated.

axioms (1)

standard math Standard properties of polynomial rings and integer-valued polynomials hold.
Invoked to establish that the linear combination preserves integrality when coefficients satisfy the special form.

pith-pipeline@v0.9.0 · 5370 in / 1197 out tokens · 24470 ms · 2026-05-15T00:41:51.731815+00:00 · methodology