Roots of generalised Hermite polynomials when both parameters are large

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 grid, as was numerically observed by Clarkson. We describe the elliptic region and the deformed grid in terms of elliptic integrals and their degenerations.