Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlev\'e III ($D_7$) equation

Abstract

We describe the use of confluent Darboux transformations for Schr\"odinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlev\'e equations. As a preliminary illustration, we briefly describe how the Yablonskii-Vorob'ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlev\'e II equation. We then proceed to apply the method to obtain the main result, namely a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlev\'e III equation of type $D_7$.