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.

arXiv comment35 pages

References in corpus (5)

• Higher order Painlevé equations of type $A_l^{(1)}$
• Determinant Structure of the Rational Solutions for the Painlevé II Equation
• On the Umemura Polynomials for the Painlevé III equation
• Transformations ${RS}_4^2(3)$ of the Ranks $\le 4$ and Algebraic Solutions of the Sixth Painlevé Equation
• A Determinant Formula for a Class of Rational Solutions of Painlevé V Equation

Cited by in corpus (1)

• From Klein to Painleve via Fourier, Laplace and Jimbo