065 Peter Miller

The Painlevé-IV equation has two families of rational solutions, which can be represented in terms of special polynomials called generalized Hermite polynomials and generalized Okamoto polynomials, respectively. The generalized Hermite polynomials have a convenient representation in terms of Hankel determinants for a suitable weight and hence can be identified with norming constants for certain pseudo-orthogonal polynomials. This connection provides a path to the analysis of the generalized Hermite rationals when the parameters are large; however it is not known whether the generalized Okamoto polynomials have a similar representation. In this talk, we explain how the isomonodromic approach places both families of rational solutions in terms of special cases of the Riemann-Hilbert inverse monodromy problem for Painlevé-IV. This allows techniques from steepest descent theory to be used to analyze both families of rational solutions within a common analytical framework.