Legendre compressions and an integrality conjecture for the Hörmander--Bernhardsson extremal function

Abstract

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander--Bernhardsson extremal function $\phi$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-λ\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,λ)\in\mathbb Z[x,λ] \qquad(n\ge0). \] 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 \[ \widetilde u_{n+1}(a^2,λ)=\binom{2n+2}{n+1}\det T_n(a,λ). \] As consequences, we derive that \[ \left(\fracπ{4C}\right)^2 \quad\text{and}\quad -\frac{L_τ(1)}{2C} \] are not simultaneously rational, where \(C\) is the sharp point-evaluation constant for $P{W}^1$, $\pm {\tau }_n$ are the nonzero zeros of $\phi$, and $L_{\tau }(1)={\sum }_{n\ge 1}\frac{(-1{)}^n}{{\tau }_n}.$ Finally, if we write $\phi (z)={\sum }_{n\ge 0}{c}_n{z}^{2n},$ then \[ c_n\in C^n\,\mathbb Z[π^2,C,L_τ(1)] \qquad(n\ge0). \]

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