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arXiv · 2503.12696

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

J. W. E. Harrow·A. N. W. Hone
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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 $D_7$.

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J. W. E. Harrow, A. N. W. Hone. 2025-03-16. Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlev\'e III ($D_7$) equation. https://arxiv.org/abs/2503.12696

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