New Coalescences for the Painlev\'e Equations

TL;DR

This paper explores the connections between Painlevé equations and Ince's equations through degeneracy procedures, revealing a cascade of relationships and extending the coalescence concept to various related equations and transformations.

Contribution

It introduces a unified degeneracy framework linking Painlevé and Ince's equations, expanding the understanding of their interrelations and transformations.

Findings

Painlevé equations connected to Ince's equations via degeneracy

Ince's equations form a cascade similar to Painlevé's coalescence

Degeneracy extended to special, symmetric equations, and Bäcklund transformations

Abstract

The Painlev\'e equations are here connected to other classes of equations with the Painlev\'e Property (Ince's equations) by the same degeneracy procedure that connects the Painlev\'e equations (coalescence). These Ince's equations here are also connected among themselves like in the traditional Painlev\'e's coalescence cascade. Such degeneracy is considered also for the special equations, symmetric equations and B\"acklund transformations.