Zeros of Large Degree Vorob'ev-Yablonski Polynomials via a Hankel Determinant Identity

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+2u3+ α. 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 [3], which used the Jimbo-Miwa Lax representation of PII equation and the asymptotic analysis thereof. © 2014 The Author(s) 2014. Published by Oxford University Press. All rights reserved.