On a class of algebraic solutions to Painlev\'e VI equation, its determinant formula and coalescence cascade

TL;DR

This paper presents a determinant formula for algebraic solutions to Painlevé VI, connecting it with universal characters and Jacobi polynomials, and explores its degeneration to solutions of related Painlevé equations.

Contribution

It introduces a new determinant formula for algebraic solutions of Painlevé VI, linking it to universal characters and Jacobi polynomials, and discusses coalescence to other Painlevé equations.

Findings

Determinant formula for algebraic solutions to Painlevé VI.

Connection between the formula and universal characters.

Degeneration to solutions of Painlevé V and III.

Abstract

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.