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 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.

arXiv comment64 pages, 20 figures, sections on the elliptic region and WKB analysis largely rewritten

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Cited by in corpus (3)

• On the algebraic solutions of the Painleve-III (D7) equation
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• Complete classification of rational solutions of $A_{2n}$-Painlevé systems