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 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.