Roots of generalised Hermite polynomials when both parameters are large

TL;DR

This paper analyzes the asymptotic distribution of roots of generalized Hermite polynomials with large parameters, revealing they densely fill a specific elliptic region and form a deformed lattice, with detailed descriptions via elliptic integrals.

Contribution

It provides a rigorous description of the roots' asymptotic distribution and organization for large parameters, confirming and extending previous numerical observations.

Findings

Roots densely fill a bounded elliptic region

Roots organize on a deformed rectangular lattice

Descriptions involve elliptic integrals and degenerations

Abstract

We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.