The Yosida Class is Universal

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We discuss families of meromorphic functions fₕ obtained from single functions f by the re-scaling process fₕ(z)=h⁻αf(h+h⁻βz) generalising Yosida's process fₕ(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 Painlevé transcendents. The Yosida class contains all limit functions of generalised Yosida functions--the Yosida class is universal.