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arXiv · 1401.1408

Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity

Marco Bertola·Thomas Bothner
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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 $\mathcal{Q}_n(x),x\in\mathbb{C}$. These polynomials are the major ingredients in the construction of rational solutions to the second Painlev\'e equation $u_{xx}=xu+2u^3+\alpha$. As an application of the new identity, we study the zero distribution of $\mathcal{Q}_n(x)$ as $n\rightarrow\infty$ 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 \cite{BM}, which used the Jimbo-Miwa Lax representation of PII equation and the asymptotical analysis thereof.

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Marco Bertola, Thomas Bothner. 2014-01-06. Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity. https://arxiv.org/abs/1401.1408

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