Discrete equations from B\"{a}cklund transformations of the fifth Painlev\'{e} equation

TL;DR

This paper derives new discrete equations from Bäcklund transformations of the fifth Painlevé equation, revealing a novel discrete equation with ternary symmetry and exploring hierarchies of rational solutions expressed via special polynomials.

Contribution

It introduces a new discrete equation with ternary symmetry derived from Bäcklund transformations of the fifth Painlevé equation and explores hierarchies of rational solutions using special polynomials.

Findings

Derived a new discrete equation with ternary symmetry.

Established hierarchies of rational solutions using Laguerre and Umemura polynomials.

Demonstrated nonuniqueness of rational solutions and their implications.

Abstract

In this paper discrete equations are derived from B\"{a}cklund transformations of the fifth Painlev\'{e} equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlev\'{e} equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is nonuniqueness of some rational solutions of the fifth Painlev\'{e} equation. Pairs of nonunique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.