Special polynomials associated with rational solutions of Painlevé equations

In this thesis, I investigate properties of special polynomials associated with rational solutions of the Painlevé equations. The Painlevé equations are second order non-linear differential equations. The Painlevé equations have a plethora of interesting mathematical properties and arise in a variety of applications such as plasma physics, random matrix theory, linear waves, and quantum gravity. The generalised Okamoto polynomials are special polynomials associated with rational solutions of the fourth Painlevé equation. The main focus of this thesis are the generalised Okamoto polynomials and polynomials that are defined in this thesis called the 4-Okamoto polynomials. Both the generalised Okamoto and the 4-Okamoto polynomials depend on several parameters. In this thesis, we investigate properties such as the structures formed by the roots of these polynomials in the complex plane and the discriminant of the polynomials. We find that various properties of the generalised Okamoto and the 4-Okamoto polynomials depend on the relative size of the parameters of each polynomial. The generalised Okamoto polynomials and the 4-Okamoto polynomials are naturally associated with families of partitions. The partitions associated with the generalised Okamoto polynomials are so-called 3-core partitions and the partitions associated with the 4-Okamoto polynomials are so-called 4-core partitions. In this thesis, we explore how aspects of partitions associated with polynomials play a role in the properties of the polynomials.