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arXiv · 1907.08552

Roots of generalised Hermite polynomials when both parameters are large

Davide Masoero·Pieter Roffelsen
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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.

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Davide Masoero, Pieter Roffelsen. 2019-07-19. Roots of generalised Hermite polynomials when both parameters are large. https://doi.org/10.1088/1361-6544/abdd93

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