Rational Solutions of the Fifth Painlev\'e Equation. Generalised Laguerre Polynomials

TL;DR

This paper explores rational solutions of the fifth Painlevé equation using generalized Laguerre polynomials, analyzing their properties, root structures, and applications, and establishing connections with partitions and discriminants.

Contribution

It introduces new classes of rational solutions expressed via generalized Laguerre polynomials and investigates their properties, root structures, and associated equations.

Findings

Two classes of rational solutions are identified.

Root structures exhibit interesting transitions and coalescences.

Discriminants are expressed in terms of partition data.

Abstract

In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlev\'e equation, the associated $\sigma$-equation and the symmetric fifth Painlev\'e system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameter. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.