An old new class of meromorphic functions

TL;DR

This paper uses the re-scaling method to analyze solutions of a Hamiltonian system related to Painleve IV, fully characterizing certain transcendents with positive Nevanlinna deficiency.

Contribution

It provides a detailed description of solutions to a recently discovered Hamiltonian system linked to Painleve IV, resolving the classification of specific transcendents.

Findings

Complete characterization of solutions with positive Nevanlinna deficiency

Detailed description of Hamiltonian system solutions

Connection to Painleve IV transcendents

Abstract

Based on the so-called re-scaling method, we will give a detailed description of the solutions to the Hamiltonian system (\ref{Hsystem}) below, which was discovered only recently by Kecker, and is strongly related to Painleve's fourth differential equation. In particular, the problem to determine those fourth Painleve transcendents with positive Nevanlinna deficiency $\delta(0,w)$, is completely resolved.