The Yosida Class is Universal

TL;DR

This paper explores the Yosida class of meromorphic functions, showing it is a universal class that includes all limit functions of generalized Yosida functions, with applications to Painlevé transcendents.

Contribution

It introduces a generalized rescaling process for meromorphic functions and proves the Yosida class's universality, encompassing all limit functions of these generalized processes.

Findings

Yosida class is universal for generalized Yosida functions.

Includes all limit functions of these functions.

Applies to Painlevé transcendents.

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.