The Yosida Class is Universal

Abstract

We discuss families of meromorphic functions $f_h$ obtained from single functions $f$ by the re-scaling process $f_h(z)=h^{-\alpha}f(h+h^{-\beta}z)$ generalising Yosida's process $f_h(z)=f(h+z)$. The main objective is to obtain information on the value distribution of the generating functions $f$ themselves. Among the most prominent generalised Yosida functions are first, second and fourth Painlev\'e transcendents. The Yosida class contains all limit functions of generalised Yosida functions--the Yosida class is universal.