An integrality phenomenon

TL;DR

This paper proves a broad result on the integrality of sequences generated by specific polynomial-coefficient recursions, including a conjecture related to the H"ormander-Bernhardsson extremal function.

Contribution

It establishes a general integrality theorem for sequences defined by recursions with polynomial coefficients, covering a conjecture in extremal function theory.

Findings

Proves a general integrality result for certain recursive sequences.

Provides a direct proof for a conjecture related to the H"ormander-Bernhardsson extremal function.

Includes a broad class of recursions with polynomial coefficients.

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.