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

Abstract

A determinant formula for a class of algebraic solutions to Painlevé VI equation (P${}_{\mathrm{VI}}$) 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${}_{\mathrm{V}}$ and P${}_{\mathrm{III}}$ is discussed by applying the coalescence procedure. Relationship between Umemura polynomials associated with P${}_{\mathrm{VI}}$ and our formula is also discussed.

35 pages