The fourth Painlevé equation and associated special polynomials

In this article rational solutions and associated polynomials for the fourth Painlevé equation are studied. These rational solutions of the fourth Painlevé equation are expressible as the logarithmic derivative of special polynomials, the Okamoto polynomials. The structure of the roots of these Okamoto polynomials is studied and it is shown that these have a highly regular structure. The properties of the Okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials. Further representations are given of the associated rational solutions in the form of determinants through Schur functions