Good linear operators and meromorphic solutions of functional equations (Q275620)

The purpose of this paper is to collect some tools of Nevanlinna theory, which are used in the study of meromorphic solutions of differential, difference, and \(q\)-difference equations, in a common toolbox for the study of general classes of functional equations by introducing the notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. Let \(\mathcal{M}\) be the field of meromorphic functions in the complex plane, and let \(\mathcal{N}\subset\mathcal{M}\). A linear operator \(L:\mathcal{N}\rightarrow\mathcal{N}\) is a good linear operator for \(\mathcal{N}\) with the exceptional set property \(\mathbb{P}\) if the following two properties hold: NEWLINENEWLINENEWLINENEWLINE (1) for any \(f\in\mathcal{N}\), \(\displaystyle m\left(r,\frac{L(f)}f\right)=o(T(r,f)) \) as \(r\to\infty\) outside of an exceptional set \(E_f\) with the property \(\mathbb{P}\); NEWLINENEWLINENEWLINENEWLINE (2) the counting functions \(N(r,f)\) and \(N(r,L(f))\) are asymptotically equivalent. NEWLINENEWLINENEWLINENEWLINE The authors apply these methods to study the growth of meromorphic solutions of the functional equation \(M(z, f) + P(z, f) = h(z)\), where \(M(z, f)\) is a linear polynomial in f and \(L(f)\), where \(L\) is good linear operator, \(P(z, f)\) is a polynomial in \(f\) with degree deg \(P\geq2\), both with small meromorphic coefficients, and \(h(z)\) is a meromorphic function. For example, let, for any \(f\in\mathcal{N}\), \(N(r,f)=o(T(r,f))\) as \(r\to\infty\) outside of an exceptional set \(E_f\) with the property \(\mathbb{P}\), and let \(\{L_k,k\in J\}\) be a finite collection of good linear operators for \(\mathcal{N}\) with the exceptional set property \(\mathbb{P}\). If \(f_1,f_2\in\mathcal{N}\) are any two meromorphic solutions of the equation NEWLINE\[NEWLINE M(z,f)+P(z,f)=h(z)\,, NEWLINE\]NEWLINE where \(P(z,f)=b_2(z)f^2+\dots+b_n(z)f^n\) is a polynomial in \(f\) with small meromorphic coefficients, \(h\in\mathcal{M}\), and \(M(z,f)\) is a linear polynomial in \(f\) and \(L_k(f)\), \(k\in J\), with small meromorphic coefficients, then NEWLINE\[NEWLINE T(r,f_2)=O(T(r,f_1))+o(T(r,f_1)), NEWLINE\]NEWLINE where \(r\to\infty\) outside of an exceptional set \(E\) with the property \(\mathbb{P}\).