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On a class of algebraic solutions to Painlevé VI equation, its determinant formula and coalescence cascade

Tetsu Masuda
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Abstract

A determinant formula for a class of algebraic solutions to Painlevé VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the Jacobi polynomials. Degeneration to the rational solutions of P$_{\rm V}$ and P$_{\rm III}$ is discussed by applying the coalescence procedure. Relationship between Umemura polynomials associated with P$_{\rm VI}$ and our formula is also discussed.

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Tetsu Masuda. 2002-02-21. On a class of algebraic solutions to Painlevé VI equation, its determinant formula and coalescence cascade. https://arxiv.org/abs/nlin/0202044

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