Zeros of Large Degree Vorob’ev–Yablonski Polynomials via a Hankel Determinant Identity

Marco Bertola, Thomas Joachim Bothner

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25 Citations (Scopus)
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Abstract

In the present paper, we derive a new Hankel determinant representation for the square of the Vorob’ev–Yablonski polynomial Qn (x),x∈C Qn(x),x∈C⁠. These polynomials are the major ingredients in the construction of rational solutions to the second Painlevé equation uxx=xu+2u3+αuxx=xu+2u3+α⁠. As an application of the new identity, we study the zero distribution of Qn(x)Qn(x) as n→∞ n→∞ by asymptotically analyzing a certain collection of (pseudo)-orthogonal polynomials connected to the aforementioned Hankel determinant. Our approach reproduces recently obtained results in the same context by Buckingham and Miller [3], which used the Jimbo–Miwa Lax representation of PII equation and the asymptotic analysis thereof.
Original languageEnglish
JournalInternational Mathematics Research Notices
Volume2015
Issue number19
Early online date1 Dec 2014
DOIs
Publication statusPublished - 2015

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