Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

In this paper, rational solutions of the fifth Painlevé equation are discussed. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalized Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalized Umemura polynomials. Both the generalized Laguerre polynomials and the generalized Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalized Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlevé equation, the associated σ-equation, and the symmetric fifth Painlevé system are expressed in terms of generalized Laguerre polynomials. Nonuniqueness 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 parameters. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviors controlled by hook data associated with the partition. The discriminants of the generalized 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.