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An integrality phenomenon

Florian Fürnsinn, Danylo Radchenko, Wadim Zudilin

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 Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.

An integrality phenomenon

Abstract

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: equals a linear combination of with polynomial coefficients in of special form. This includes a conjectural integrality of the sequence related to the Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.

Paper Structure

This paper contains 2 sections, 3 theorems, 27 equations.

Table of Contents

  1. An irregular example of integrality
  2. A general integrality statement

Key Result

Theorem 1

The sequence $u_n$ generated by the recursion u-rec1 and initial data $u_0=1$, $u_{-1}=0$, is integer-valued: $u_n\in\mathbb Z[b,c]$.

Theorems & Definitions (6)

  • Theorem 1
  • proof : Proof of \ref{['ex-conj']}
  • Theorem 2
  • Theorem 3
  • proof : Proof of \ref{['thmrec']}
  • proof : Proof of \ref{['thmpoly']}