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 language | English |
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Journal | International Mathematics Research Notices |
Volume | 2015 |
Issue number | 19 |
Early online date | 1 Dec 2014 |
DOIs | |
Publication status | Published - 2015 |