On the product of the extreme zeros of Laguerre polynomials

TL;DR

This paper investigates a novel property of the product of extreme zeros of Laguerre polynomials, linking it to a longstanding conjecture for Hermite polynomials and advancing numerical methods in quantum mechanics.

Contribution

It uncovers an unknown property of Laguerre polynomial zeros and connects it to a 20-year-old Hermite polynomial conjecture, using a parametric eigenvalue approach.

Findings

Identifies a new property of Laguerre polynomial zeros.

Relates the property to a conjecture for Hermite polynomials.

Develops numerical methods relevant to quantum mechanics.

Abstract

The purpose of this note is twofold: firstly, it intends to bring to light an apparently unknown property of the product of the extreme zeros of Laguerre polynomials, which in a very particular case leads to a twenty-year-old conjecture for Hermite polynomials posed by Gazeau, Josse-Michaux, and Moncea while developing numerical methods in quantum mechanics; and secondly to progress towards the solution of this problem as an application of a parametric eigenvalue problem.