PreprintCC0arXiv
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 Lm(α) of degree m and parameter α>−1, the product of its jth smallest and jth largest zeros is strictly increasing with m for each fixed positive integer j, once m≥2j−1. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-N Hermite polynomial HN, in both parities and for every N≥4. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.
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BibTeX
@misc{castillo2026symmetric,
title = {{Symmetric products of Laguerre zeros}},
author = {K. Castillo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08480},
eprint = {2609.08480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08480},
note = {CC0},
}
RIS
TY - UNPB
TI - Symmetric products of Laguerre zeros
AU - K. Castillo
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08480
UR - https://arxiv.org/abs/2609.08480
AB - 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 jth smallest and jth largest zeros is strictly increasing with m for each fixed positive integer j, once m≥2j−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≥4. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.
ER -