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

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

2002-02-21

arxiv.org/abs/nlin/0202044v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

35 pages

Scientific paper

A determinant formula for a class of algebraic solutions to Painlev\'e 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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