On functional equations for meromorphic functions and applications
Let \(P\) and \(Q\) be polynomials. The authors consider a functional equation of the form \(P(f)=Q(g)\), where \(f\), \(g\) are meromorphic functions in the complex plane. In terms of Nevanlinna theory, the research has been continued since [\textit{G. Barsegian} (ed.) et al., Value distribution theory and related topics. Boston, MA: Kluwer Academic Publishers (2004; Zbl 1081.30001)]. They call a polynomial of the form \(z^n+az^{n-m}+b\) Yi's polynomial, where \(a\) and \(b\) are non-zero complex numbers. This type of polynomial possesses connections with the study of the uniqueness problem for meromorphic functions. One of the main results for the functional equation \(P(f)=Q(g)\) with Yi's polynomial is the following. Let \(n\), \(m\) be positive integers, \(c\), \(d\), \(e\), \(u\), \(v\), \(t\in\mathbb C\) be non-zero constants, and let \(n\geq 2m+4\), and either \(m\geq 2\), \((n,m)=1\), or \(m\geq 4\). Suppose that \((f,g)\) is a non-constant meromorphic solution of the equation \[ cf^n+df^{n-m}+e=ug^n+vg^{n-m}+t. \] Then there exists a non-zero constant \(h\) such that \(g=hf\). The authors also discuss an analogue of Ritt's decomposition theorem for a class of polynomials of Fermat-Waring type in meromorphic functions.
scientific article; zbMATH DE number 6813953
scientific article