Legendre compressions and an integrality conjecture for the H\"ormander--Bernhardsson extremal function

TL;DR

This paper proves a conjecture related to the integrality of a three-term recurrence for the H"ormander--Bernhardsson extremal function, using determinant comparisons in Legendre basis, with implications for rationality and coefficient properties.

Contribution

It establishes the integrality of the recurrence coefficients and connects them to tridiagonal matrix determinants, proving a specific conjecture in extremal function theory.

Findings

The recurrence coefficients are polynomials with integer coefficients.

Certain constants related to the extremal function are proven to be irrational.

The coefficients of the extremal function are contained in a specific algebraic ring.

Abstract

We prove Conjecture~2 of Bondarenko, Ortega-Cerd\`a, Radchenko, and Seip for the three-term recurrence attached to the H\"ormander--Bernhardsson extremal function $\varphi$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-\lambda\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,\lambda)\in\mathbb Z[x,\lambda] \qquad(n\ge0). \] The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,\lambda)$ denotes the $n$th BOCRS tridiagonal truncation, then \[ \widetilde u_{n+1}(a^2,\lambda)=\binom{2n+2}{n+1}\det T_n(a,\lambda). \] As consequences, we derive that \[ \left(\frac{\pi}{4C}\right)^2 \quad\text{and}\quad -\frac{L_\tau(1)}{2C} \] are not simultaneously rational, where \(C\) is the sharp point-evaluation constant for $PW^1$, $\pm\tau_n$ are the nonzero zeros of $\varphi$, and $ L_\tau(1)=\sum_{n\ge1}\frac{(-1)^n}{\tau_n}.$ Finally, if we write $\varphi(z)=\sum_{n\ge0}c_n z^{2n},$ then \[ c_n\in C^n\,\mathbb Z[\pi^2,C,L_\tau(1)] \qquad(n\ge0). \]