发表机构
University of Coimbra（科英布拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明并推广了关于Hermite多项式极端正零点的猜想，通过广义Laguerre多项式零点对称积的单调性，并解决了一个单参数扩展。
AI 中文摘要
我们证明并实质性地推广了Gazeau、Josse-Michaux和Monceau关于Hermite多项式极端正零点的猜想，该猜想源于相平面的有限维量子化。对于次数为$m$、参数$\alpha>-1$的广义Laguerre多项式$L_m^{(\alpha)}$，其第$j$小和第$j$大零点的乘积，在固定正整数$j$且$m\ge2j-1$时，随$m$严格递增。经典的Hermite-Laguerre恒等式随后证明了次数为$N$的Hermite多项式$H_N$的原始猜想，涵盖两种奇偶性且对所有$N\ge4$成立。作为进一步推论，涉及相应Jacobi矩阵形变的一个单参数扩展也得到了解决。
英文摘要
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 $L_m^{(α)}$ of degree $m$ and parameter $α>-1$, 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.
