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

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 Painlevé 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 $n\to \infty$ 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.

arXiv comment36 pages, 18 figures