Symmetric products of Laguerre zeros

Abstract

We prove and substantially extend a conjecture of Gazeau, Josse-Michaux, and Monceau concerning the extreme positive zeros of Hermite polynomials, which arose from a finite-dimensional quantisation of the phase plane. For the generalised Laguerre polynomial $L_m^{(\alpha)}$ of degree $m$ and parameter $\alpha>-1$, the product of its $j$th smallest and $j$th largest zeros is strictly increasing with $m$ for each fixed positive integer $j$, once $m\ge2j-1$. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-$N$ Hermite polynomial $H_N$, in both parities and for every $N\ge4$. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.