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

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 \[ c_n\in C^n\,\mathbb Z[\pi^2,C,L_\tau(1)] \qquad(n\ge0). \]