Hankel Determinant Approach to Generalized Vorob'ev-Yablonski Polynomials and their Roots

TL;DR

This paper introduces Hankel determinant identities for generalized Vorob'ev-Yablonski polynomials, enabling analysis of their roots and boundary patterns, advancing understanding of solutions to the Painlevé hierarchy.

Contribution

It provides new Hankel determinant formulas for these polynomials and applies them to analyze root patterns and boundary curves.

Findings

Derived Hankel determinant identities in terms of Schur polynomials.

Analyzed root distributions and boundary curves of the polynomials.

Connected polynomial identities to numerical root pattern observations.

Abstract

Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlev\'e hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob'ev-Yablonski polynomials and provide formul\ae\, for the boundary curves of the highly regular patterns observed numerically in \cite{CM}.