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Fixed points of meromorphic solutions for difference Riccati equation. (English) Zbl 1280.30015

In this paper, the authors treat meromorphic solutions of the difference Riccati equation \[ f(z+1)=\frac{A(z)+\delta f(z)}{\delta-f(z)},\tag{1} \] where \(A(z)\) is a non-constant rational function, and is not a polynomial. They consider fixed points of shifts and differences of meromorphic solutions of (1) when \(\delta=\pm1\) and \(\deg A=2\). For a meromorphic function \(f(z)\), they define \[ \tau(f)=\limsup_{r\to\infty}\frac{\log N(r,\frac{1}{f-z})}{\log r}. \] They show that every finite order transcendental meromorphic solution of (1) satisfies (i) \(\tau(f(z+n))=\sigma(f(z))\), \(n=1, 2, \dots \); (ii) if there is a rational function \(m(z)\) satisfying \[ m(z)^2=\left(\frac{z}{1+z}\right)^2-\frac{4A(z)}{1+z}, \] then \(\tau(\frac{\Delta f(z)}{f(z)})=\sigma(f(z))\); (iii) if there is a rational function \(n(z)\) satisfying \(n(z)^2=z^2-4A(z)\), then \(\tau(\Delta f(z))=\sigma(f(z))\). The condition “finite order” is required. In general, (i) does not always hold for infinite order meromorphic functions.
This work is a continuation of [the second author and K. H. Shon, Acta Math. Sin., Engl. Ser. 27, No. 6, 1091–1100 (2011; Zbl 1218.30076)].

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B12 Iteration theory, iterative and composite equations

Citations:

Zbl 1218.30076
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