On generalized Dhombres equations with non-constant rational solutions in the complex plane
Let \(w_0 \in \mathbb{C}\) and \(r > 0\). The authors characterize all pairs \((f,\varphi)\) of non-constant rational functions \(f\) holomorphic at zero with \(f(0) = w_0\) and of non-constant meromorphic functions \(\varphi\) holomorphic at \(w_0\) such that \[ f(zf(z)) = \varphi(f(z)) \tag{*} \] for \(|z| < r\). The following theorem is proved. Let \(w_0\) be a primitive root of 1 of order at least 2 and let \(\varphi_0\) be a non-constant meromorphic function holomorphic at \(w_0\). (i) If \(\varphi_0\) is different from \(w \mapsto w_0(1 + w_0 - w)^{-1}\), then all formal solutions \(F\), \(F(z) = w_0 + a_m z^m + \cdots \) of the equation \[ F(zF(z)) = \varphi_0(F(z))\tag{**} \] are given by Taylor expansions of the rational solutions of the equation \[ f(zf(z)) = \varphi_0(f(z)). \] (ii) If \(\varphi_0(w) = w_0(1 + w_0 - w)^{-1}\), then (*) has non-constant rational solutions \(f\) with \(f(0) = w_0\). But there exist local analytic solutions \(f\) with \(f(0) = w_0\) of (*) and non-convergent formal solutions \(F\) of (**) with \(F(0) = w_0\) which are not Taylor expansions of rational solutions of (*).
scientific article; zbMATH DE number 6128638