Classical transcendental solutions of the Painlev\'e equations and their degeneration

TL;DR

This paper provides determinant formulas for classical transcendental solutions of Painlevé V and VI equations and explores how these solutions degenerate through coalescence processes.

Contribution

It introduces explicit determinant expressions for solutions and analyzes their degeneration during Painlevé equation coalescence.

Findings

Determinant expressions for Painlevé V and VI solutions

Analysis of solution degeneration via coalescence

Explicit formulas enhance understanding of Painlevé transcendents

Abstract

We present a determinant expression for a family of classical transcendental solutions of the Painlev\'e V and the Painlev\'e VI equation. Degeneration of these solutions along the process of coalescence for the Painlev\'e equations is discussed.